Should I use "As" or "Since" in this sentence?

You are right. Since is a possible answer here, meaning because, just as you suggest.

However, if the instruction or task is to choose the "best" answer, then as is "better".

The sentence uses the phrase more and more, which suggests a process of increasing over time. As can mean while so is more likely in the sentence.

If we began with Since, meaning because, we'd more likely leave out more and more or say something like Since he exercised more and more, his muscles became more and more developed.

Note: In any case, learning facts like "differences between as and since is a very inefficient way to improve English proficiency. All possible such rules in English are endless in number and complexity. Reading more things you like at a level that's easy for you is much better.