Well like you said "most will accept proof that I submitted and are waiting on the Document", but I will no doubt get the one USCG or local law enforcement officer that won't. That is just my luck . Especially traveling thru 9 states without State registration numbers or stickers on display. And at this point, since I have already sent in my forms and with an on going Gov shut down, I don't know if a priority request will do me much good, but it is worth checking out.
So the boat did not have an old name, hailing port, or any reg stickers on it? If it does, I would leave the existing on until your stuff is in order. If its blank, i would add your name and hailing port as to not attract unneeded attention. LEO will see FL as the hailing port and likely no one will bother you about missing FL stickers on your trip down to FL... I have NEVER heard anyone having issue so long as they could provide proof they submitted the doc.
If you are traveling in winter, spring, I would not worry too much. I saw practically zero LEO out on the water during my 3,000 mile trip from Jan til May. I have purchased many boats before with documentation that took 5-6+ months to get and had no issues. I would not be worrying about it just keep proof of the paperwork that you submitted on-board.
And to reiterate, a doc service/agent does not magically help with a backlogged documentation that has yet to be processed by the CG. In fact my experience is they can mess things up and delay things more. Their letter does nothing magical. In order to get the priority request processed, you will need a reason such as a trip to the Bahamas. If you happen not to take that trip then so be it but you need a legit reason. Again, I don't think you need the documentation processed priority for the purposes that you have stated. One thing that would worry me is being in a state that does collect sales/use tax and having to pay them and FL at the same time. Be sure to read the regs of the state where the boat currently is and see if it might be in your interest to relocate the boat etc.