Did биток always only exist in the context of the idiom "битком набиты", or was it a "perfectly valid noun in its own right" at some point in the past?
Though the word Биток exists as a separate word (see below), this particular saying (набить битком) is similar to others like "бежать бегом" "сидеть сиднем", "волочь волоком" (see more details here).
This form of word is tautological enforcement of the idea of packing something very tightly. "Бежать бегом" means "run very fast", "сидеть сиднем" - sit without moving, "волочь волоком" - drag something very heavy without any other possible option to move it.
Below is a meaning of the word Биток itself, but all these meanings have nothing to do with your original saying:
I м. разг. Небольшая круглая котлета из рубленого мяса; биточек.
II м. разг. Предмет (палка, бабка и т.п.), которым бьют при игре в городки, бабки и т.п.;
"perfectly valid noun in its own right"
[битком] in [битком набиты] is not a noun. It is adverb here.
Noun is a grammar part, which answers question: what? by what?
Adverb is a grammar part, which answes question: how?
[битком набиты] is not [filled by bitok]
[битком набиты] is [filled in bitok-style] or [filled bitokly].
Did биток always only exist in the context of the idiom "битком набиты"
You can not use [биток] on its own as a noun when you want to say, that room is overcrowded. For example, you can not say there is [биток] in the room. Therefore, when speaking about crowed room, this word was never used as standalone.
[биток] can be used as a noun, when it is a piece, a thing, something, that you can take in your hand.
"Биток" - is also a white ball in billiard. The only one you allowed to strike. The one that puts every other ball in the basket. So "Битком набиты" could have come from billiard.