There are many tire diameter caluclators on the internet google is your friend. You can see the actual differences that way, and make a judgement if there will be clearance. I think you have to be pretty far off in outer diameter if you stay with 13 inch many will probably fit fine but it's worth considering so check it out.
The other thing is load rating, make sure those new ones are rated appropriately for the weight. Most real trailer tires should be maintained at 80psi, so they're a much different animal than any car tire.
Radial type trailer tires are fine, less rolling resistance so they run a little cooler but they must be rated for trailer use.