Word for the thing that moves back and forth in a watch mechanism

I believe that would be the escapement.

Here’s a description:

The escapement is a mechanism that translates rotational energy into lateral impulses. The tick-tock sound you hear when holding a watch to your ear is from the escapement. The pallet fork locks and unlocks with the escape wheel at each vibration of the balance wheel.
_{Source: Hodinkee — Watch 101}

In real life, though, few people would have any idea of what you’re talking about.

I submit to you instead a mashup that employs a few productive English language tactics:

• onomatopoeia (the process of creating a word that phonetically imitates, resembles, or suggests the sound that it describes)

• metonomy (a figure of speech in which a thing or concept is referred to by the name of something closely associated with that thing or concept)

• derivational suffixing with -er (turning a verb into noun that is person or thing that does the verb).

So, here you go . . .

tick-tocker

Here’s an example usage:

I can’t hear anything; the tick-tocker must be broken.

Proceed at your own risk.

Just balance will do.

balance

15. Also called balance wheel. Horology. a wheel that oscillates against the tension of a hairspring to regulate the beats of a watch or clock.

https://www.dictionary.com/browse/balance