Jimmy - based on all the information here, especially that there are multiple bad guys with the same name, if it were me I'd have included my SSN on the 4473. If there is any bad news that would result, IMO you don't want that to happen later on instead of now. If there is a hiccup, you might be able to talk to an attorney to get it cleared up.
Unfortunately, you can't even get a FL drivers license anymore without giving them your SSN. Identity theft doesn't seem to be the greater of the potential problems in this case.