I think it depends. I've seen lots of cars that are truly antiques and registered as such w/ a trailer behind them going to car shows and the like. The DMV doesn't give them issues that I've heard of because they are probably reasonable enough to know there is generally just a few lawn chairs and a cooler in there which I wouldn't constitute as cargo any more than a pair of sunglasses or a pack of cigarettes which of course is perfectly legal.
However, I don't know how it is in your part of the state but around here I wouldn't even think about overloading a trailer or my truck beyond what the registration calls for right now. W/ the oil boom due to the marcellus shale cops have been on every loaded vehicle like stink on poo since they are trying to crack down on the oil guys who are carrying very large loads(they have to hit everyone to be fair).
I used to load my f-150 to the gills w/ firewood but don't dare anymore as it's a hefty fine if I get checked and my truck is overloaded or I go over my weight class 2 registration classification. 3 years ago, nobody would have blinked an eye so long as the bumper wasn't digging up the asphalt.
In your case however, I'm sure the firewood is a No, No by the statement you referenced.
If you want to do this, get the truck registered as a current vehicle. As a classic you have to get it inspected annually anyway so the only difference is you'd have to pay your yearly registration right? I believe my class 2 registration just cost me 81 bucks. The only thing would be if you have emissions testing in your area.
Here is the penndot fact sheet regarding antique and classic plates
http://www.dmv.state.pa.us/pdotforms/fact_sheets/fs-ant.pdf
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