Is a bigger trailer better? Yes
Would your trailer work? Yes
The trailer may only be rated at 3500 lbs and that doesn't include the weight of the trailer (maybe 400 lbs) but no one on this forum seems to understand TONGUE WEIGHT. Weight that is NOT on the trailer and instead on the truck.
4000 lb tractor on 400 lb trailer with typical 20% tongue weight = 3520 lbs on the trailer total. Certainly not greatly exceeding the limits and with a single axle you can easily obtain 25-30% tongue weight with the proper hitch. (3,300-3,080 lbs on trailer)
Once again:
Is a bigger trailer better? Yes
Would your trailer work? Yes!
For the money, I too would rent the trailer you got for that 600 mile trip but I would have no problem using your trailer for a trip across town.