I do not think there are any 13" tires that will do the job you want them to do. So moving up in size is the only realistic option.
I would move up as much as possible. Not seeing your setup, I cannot judge what that is. But it will cost similar in rims etc. to move up, so why go part way only to find you should have went all the way? (voice of experience here)
Tires get better as you go up in size until you reach 16" which is tire nirvana. Many different tires at all price points in that rim size.
If you like your TT enough to make the necessary changes to make 16s fit, you will not be disappointed.