Assuming you can find 14" wheels with the same bolt pattern and 0 offset, I'd go that route. You should be able to fit them, but just to be sure, an ST205/75/R14 load range C has a diameter of 26.1"- so make sure you can fit two of them in there with some room for the axles to move. You'll need to basically jack up one axle to the point of lifting the trailer off the ground and then measure the distance between the hub and top of fender. That needs to be about 13" or more. This shouldn't be a problem because it looks like the average B78-13/ ST175/80R13 is about 24" in diameter- so unless you're extremely close right now, it should work. If you wanted to try to do 15", the ST/205/75/R15C is 27.1, so you'd need 13.5-14" to clear.
You should also check width, because I know your B78-13's are narrower than an ST 14. The 205 has a width of about 8.5", so you need to at least fit a piece of paper up in that fenderwell in order for it to work. Ideally to avoid rubbing on turns, you should have more like 10", IMO. I couldn't find a width for the 13", but I do know they are much narrower.
Even though I think the 14" trailer tires can be found in load range B, if you're going through the trouble of upgrading, I'd just go with LRC. You have to go with C in the 15". Just be cautious to choose the 205, not a 215 or 225. The 205's give you plenty of height without getting into a wider tire that could give you width fitment problems.
However, all this said, you can see there isn't much height difference between the tires- the most you'd fit in there is a 205/75/R15, if that (because it's an older trailer designed for the smaller tires), but still you're talking about a 3" lift at the most. The Dexter over-under lift kit provides about 4" of lift, so it is close to doing that, but not more.
2007 T-286SR Cherry/Granola, #6236, original owner, current mileage: 9467.8 (as of 5/26/19)
1997 T-2653 Blue Denim, #5471
1979 12 1/2' MC, Beige & Avocado, #4639
Past Sunlines: '97 T-2653 #5089, '94 T-2251, '86 T-1550, '94 T-2363, '98 T-270SR