Checking that axle rating is your first concern. Then go to a certified weigh station; truck stop, salvage yard, etc., and have the trailer weighed empty. Then you will have some idea what you can haul.
Based on the average of 13 ft trailers as listed in "Trailer Weights in the Real World", spreadsheet available here:
you should be good for 1700 to 1800 lbs, provided the axle is in good shape. I did not include that 800 lb Burro listed, as that number is too far off the average to be trusted.
When you changed the tires
, was the wear pattern on the old tires even? Uneven wear can be a sign of a misaligned or worn axle.
If you put ST (Special Trailer) tires on it, you want to run them at the max rated pressure listed on the sidewall.
I don't know how much 'stuff' you guys have, but if it's significant you may do better to rent a U-haul
or some such van and make your move in two vehicles, the van with most of the household stuff and the rest in the 4 Runner and the trailer. So long as you follow the 4 Runner owner's manual regarding tow weight, tongue weight (10-15 % of trailer weight) and Gross Combined Vehicle Weight, you might be able to do it without the rental if you guys don't have a bunch. If you live in a furnished apartment you may be able to do it.
If you decide you can do it in the Runner and the Burro, load it up and get the whole thing weighed again, just to be on the safe side.
Good luck with your decision.