The short answer
The short answer is “no.” Russian writers do not appear to capitalize a word for a family member even when it is used as a name. This is true whether directly addressing the family member or not (or adding a modifier like "little/big" or "younger/older"), and true whether or not it is translated into English as "my brother" or simply "brother." Here are some examples from Reverso:
"We're just trying to have a peaceful drink over here, brother."
"Мы тут пытается спокойно посидеть, брат."
"We've been through a lot, big brother."
"Мы прошли через многое, большой брат."
"I will bring you back, my brother, I promise."
"Я воскрешу тебя, брат, обещаю."
"... while my brother is eating corpses, he is not my brother"
"... "'пока брат ест трупы, он мне не брат', ..."
This last example is not from Reverso but from an informБЮРО online article titled, "Уехавшая в Одессу жительница Акмолинской области называет свою мать трупоедкой."
The only one legitimate exception I can think of, where "брат" is capitalized in such instances are those in which "brother" begins the sentence as in:
"Brother threatened to cut off my ears so I'd never hear Chopin again."
"Брат угрожал отрезать мне уши, чтобы я никогда не слушала больше Шопена."
But you may also find other exceptions as in:
"Bless me, Brother, for I have acted against the example of the Gods."
"Благословите меня, Брат, ибо я поступал вопреки наставлениям Богов.
But this last example seems to be the exception and not a rule/standard/common practice and you're bound to find others similar to it.
Other forms of address: honorifics
I also took a look at what examples could be found via a Google search and was immediately reminded that, in English, we sometimes capitalize the word “brother” when referring to monks as in the following example:
“Brother Kevin looped his arm around Andrew’s neck, while Brother Joshua quickly followed.”
By the way, the example above comes from a book titled, Enclosure: A Trappist Tale. I have only read a few lines of it and, well, let’s just say it goes beyond what most might consider “brotherly love” and very much ventures into what some, if not many, would consider racy, progressive (and/or even seditious), but it provides a good example of what I’m trying to teach here linguistically.
“Jack sat up, interested now, while Brother lay down and put one arm over his eyes.”
—From Brother, Brother, p. 14
My cursory search on Russian monks did not rule out that they, too, are not also referred to as “brothers” and addressed as “Brother __________” (or “Брат __________”), but I did not quickly find any evidence that they are addressed or referred to in this way. After doing some searches with "сестра," however (see below), I suspect they can be (even though I haven't always seen "брат" capitalized in such instances). I would think that there is some sort of standard protocol on such a thing, but my access to authoritative sources on the Russian language is somewhat limited due to various factors (a language barrier for starters).
Be that as it may, in English, this use of “brother” to refer to a member of a religious order can also be seen with its female counterpart — sister— and can extend beyond the honorifics used for traditional houses of religion. If the following example is any indicator, this appears to be true for the Russian language as well.
"Yes, put those away before Sister Constance sees you."
"Спрячь это, пока Сестра Констанция не увидела."
I can’t be 100% certain because I’ve never seen the show and am relying on a Wikia FANDOM page here, but Sister Constance appears to be a nun who occasionally appears in a show called Orange is the New Black.
Other, less traditionally religious examples follow:
"And will be as long as the Sister of the Dark lives."
"И будет пока Сестра Тьмы жива."
"Они жили тихой жизнью пока Сестры света не похитители ее."
"They were living a quiet life until the Sisters of the Light abducted her."
A capitalized “сестра” can also refer to what is known in British English as “a senior female nurse, typically in charge of a ward,” as in the following example:
"I thought they were rabid until Nurse Crane filled me in."
"Я думал, это бешенство, пока сестра Крейн не просветила меня.
The fraternity kind of brother
I’ve also read that Brother, when capitalized, can also refer to a member of a fraternity, but having met some fraternity brothers in my college days, I would have to say that I believe this usage is archaic if not dying out. Although, I do believe I’ve seen a couple of books where this appeared to be a part of the dialogue. However, it seemed that this was something that might have been in use at one time at older, established, private universities such as one of the Ivy Leagues. I could be wrong, but I don’t recall any of the fraternity brothers I met ever addressing one another as “Brother ____________.”
A bit on vocatives
In Russian, siblings (and I assume other family members and the like) are not capitalized even when used as names and even when used as a vocative (unless the vocative is used in an exclamatory way or starts the sentence/phrase/expression). This finding was limited to what Google’s Ngram and search of Russian web pages indicates where I only tested this with “брате,” but I suspect similar results would be found for other family members. Furthermore, some web pages I encountered seem to back this up:
“Recall also Pushkin’s «Что тебе надобно, старче?» … (‘What do you need, old man?’) …”
—From "Is There a Vocative Case in Russian?"
Odds and ends regarding the capitalization of words for family members
In English, you might also capitalize a family member in circumstances where there is more than one and you are referring to them by number or letter (e.g., Brother #4 or Brother D). I’m not entirely sure one would do that in Russian and my guess is that common practice would be to keep the word lowercase.
With or without the preceding examples, I realize that some native English speakers may not fully be aware of all the many conventions for capitalizing (or not) one’s various family members. For a decent discussion thread that goes into somewhat more depth on this, I recommend the following from the English Language & Usage StackExchange:
Capitalization of “sister” and “brother”
For cultural aspects regarding the use of the word “brother,” you may want to visit this one also:
Use of “brother” in non-family and non-religious contexts
It doesn’t deal with the issue of capitalization but touches on other aspects that may or may not be relevant to the Russian language. Regardless, both Russian and English speakers, of which this forum may have several, might benefit from reading it.
The following is something of an aside, but I felt it relevant enough to add here:
I mentioned in my question that I also sometimes see “big brother” capitalized in Russian. This is true even when it has not been in English (and typically wouldn’t be when not used as a name). This is true whether referring to one’s older brother or a surveillance state. Keep in mind that the examples that follow these next ones (and the ones that preceded it) are direct copy from Reverso and stray from the literal a bit here and there even when it doesn’t seem necessary to do so. I found this to be true of the next couple of examples, especially:
"As all you know, George is my big brother."
"Как вы знаете, Джордж - мой Старший Брат.
"You saw watchtower as big brother and you turned away."
"Ты думал, что Сторожевая Башня это Большой Брат, поэтому и отвернулся."
But even then, there are exceptions as in:
"I could be a Big Brother."
"Я могу быть 'старшим братом'".
To recap briefly, family members are not capitalized in Russian even if being used as a name for someone and not a noun. In English, it would be technically more correct to write, “While Brother reads” rather than “While brother reads,” but you could conceivably see examples of both. An example of correct usage in English (again, from Reverso) is as follows:
We don't have the time to wait for Brother's graduation."
**Нам некогда ждать, пока брат окончит школу.
What the relative prevalence is of instances where possession is implied and where it is not, I could not tell you. In fact, I do see a lot of examples on Reverso where a possessive pronoun is used before a word denoting a family member, so if you use it, you don't run the risk of not being understood and it is unlikely that it will even sound odd to a native Russian. Having said that, it does appear that this lack of a possessive pronoun seems to occur more when the noun representing the family member comes after the following words:
or the following prepositions:
за, к, с
or immediately follows a verb.*
*It looks as if the possessive pronoun might be more likely to be dropped in instances where the word for the family member is in the dative, prepositional, and/or quite possibly the instrumental.
But these are just observations from what I've seen at Reverso, and, for the sake of full disclosure, mainly based off of the word "сестра," so I can't be certain that this observation is true in general.
Hope that helps someone out there who might have a similar or related question and if you are a native or fluent Russian speaker or teach it to others and you have something to add or disagree with anything I’ve written, please post a comment.