First off, you do NOT need 10/3. If you want, or need, a 30 amp circuit then 10/2 is typical for this type of circuit.
If all you are feeding is this heater, and even another (up to 3840 watts) then 12/2 is perfectly fine. The #12 will be MUCH easier to work with, believe me.
Do you have enough room? Only you can tell.
Are there any open breaker spaces? If not can you install twin breakers to make some room? The panel label can tell you if your panel will accept twin, or tandem, breakers.
Please post back and let us know how it turns out....