I own an 84 dx1300. Can confirm it's a 5 speed transmission with tall gearing. The trans is out right now being reconditioned for re-installation.
Here's a build link
1984 crx project - GasSavers.org - Helping You Save at the Pump Hypermiling and Fuel Efficiency Forum
Many tire shops carry inventories of used tires in good condition for cheap. Check around. Your gearing is taller than the si transmission, but would be improved with a 84 dx1300 transmission if you can find one.
Some useful info:
CRX Specs & Performance
These transmissions only weigh about 70lbs, so it can be swapped in a driveway with jack stands. Keep a look out on car-part.com! If you can get the money together for really good tires, the Michelin Destiny is a good choice. The best part is that all the tires in this size are very light to begin with. There would be less benefit to a LRR 13" due to this fact. Higher rated sidewall pressures are a good specification to look for.