Your trailer has a dry weight of 5,035 pounds. If your truck is a V8-powered 2055 Sierra, I can't see why it wouldn't.
Even if you had the 4.3 litre V6 version a well-rigged hitch should do a good job on that light a trailer, as long as you weren't going to also carry a load of stone or a couple of Harleys in the box.
Just keep in mind as you do the rebuild on your trailer that you do not want to add a lot of weight. Marble floors and a granite counter for example, go well beyond what is necessary or desirable.
Just as a side note, I tow my '74 31-foot Sovereign
(same weight as yours) with my wife's 2007 Ford Freestar. We put 4,000 miles on it last fall, and it pulled nicely.
Enjoy your time putting the trailer back in shape! Show us pictures, please?