There is a form for transferring SBRs and Suppressors, you just don't list a return date, or leave that space blank. I think it's ATF 5320.20.
I also recommend HeavyD. Cheap insurance, y'know?
Fill out the Form 5320.20, check no for number 2, and send it in. Regarding your trust the one you have will most likely be fine in FL. Who did it or is DIY? And I'm pretty sure there are no requirements to file a trust with a city or county in Florida.