Generally speaking, with an LLC, the truck is titled & registered in the LLC name, along with the commercial insurance policy also in the LLC name, then all filed with the FMCSA under the LLC name to obtain your authority.
Not seeing what all you got going on, it sounds like the problem is with two different names with the FMSCA. They want everything in one name, exactly written, including address. It all has to match.
As @God prefers Diesels suggested, you could do a "lease" to your LLC. Another solution would be to file a DBA, while maintaining your personal name on the vehicles, insurance policy, and your authority application. Another solution could be to apply for a new vehicle title in the LLC name. As far as "refinancing" goes, not sure how that plays into it. In a nutshell, all the names gotta match up or it's rejected.