I just did a more extensive search for 13" tires. Looks like Carlisle still has 175/80R13s in both C1 and D1 load ratings. Heartland also has that tire, but only in the C1 load rating. Same for Taskmaster and Hankook. I can't find other brands in that size, and you DO want trailer tires on a trailer, not automobile tires, because you need the extra sidewall stiffness.
I don't have any experience with any of these, but they are available through Discount Tire and would probably be safer than the tires currently on the trailer. I've read mixed reviews about Carlisles, but I'd probably choose them in the D1 rating out of the choices I've seen, just because of the increased load rating. Remember, these trailer tires are all speed limited to 65 mph. Don't go faster than that because they will heat up too much.
- Jack