Can the at-fault driver’s insurance company refuse to compensate me because I was driving without insurance?
No, they can’t. You should carry the insurance required by Florida law but stuff happens. In the event that you don’t have insurance, you can still make a third party claim. The third party carrier will be entitled to certain setons because you don’t have the coverage that you need, that you should have under Florida law.
But the answer to your question is no.