If I get bids lower than what my insurance company paid, can I keep the difference?

It all depends on how you are paid. In rare cases, you could keep the difference. Here’s an example:

The insurance company paid you in full, less your deductible, and the amount you owe for the work performed was less than the insurance payment. This is extremely rare and may be because the work performed was less than what the insurance company paid for or expected would be done.

In almost all other cases, the answer is no, because of the way the insurance company makes the payments: ACV vs. RCV.