So I’m going to give you a bit of info to mull over, and while some will call this nit picky, others myself included call this making informed decisions. First, the trailer you linked I agree is over priced. I did the currency conversion and my trailer (8x16 tandem axle 6k brakes load ramps spare tire was right around 200$ more), so find a better deal or as suggested rent a few and trial them. Next, I re read your original post, a cord at a time, white birch or ash, approx 30 kms on the highway. Now, your truck is rated for it (assuming payload capacity is there also, remember 10-15% tongue weight etc). Is this wood going to be wet or dry? I’d imagine safe to assume if in log form it won’t be dry. According to internet birch weighs in green at 4312 lbs, ash at 4184 lbs. The link your provided has the payload of the trailer at 4110 lbs, so if hauling green wood your technically overweight.
Im not sure how it is for you in your area, but for example, here being on the highway or interstate opens you up for potential trouble (ie law enforcement stops you, weighs you, finds your trailer over weight); or god forbid you get into an accident and they find your over weight, now you could be held liable etc.
Now, is it likely you’ll be stopped by police etc? Probabaly not. Is it likely you’d get into a wreck with that short drive? Probabaly not. But they are possibilities, and therefore risks associated with it that you should know, and take into consideration making your decision.
Factor in perhaps you get different species of wood at times etc, as some pointed out the dry weight of some species alone would max, or exceed that trailer. The atv obviously wouldn’t be a problem. Just want you to think about these things, so you can make an informed decision/best decision for you regarding doing this safely and correctly. Sorry for the long post.