Is it possible to reinstall Mountain Lion using Lion Time Machine backup?
When a newer Mac migrates data from an older OS, the system has upgrade scripts to make sure that the data lands properly on the new system.

These scripts handle all Apple software and perhaps a few high profile programs in case that Apple has found a need to clean up other data.

Migration the other way isn't tested, no scripts or support is provided and in fact, the system normally detects you are moving data backwards and refuses to restore in the first place.

You should be fine restoring a Lion backup onto any Mac with Mountain Lion. You don't need to reinstall anything - just don't run the Setup Assistant until you have the external drive and can move everything to the new mac.

Running an older OS on a newer Mac is also problematic, since often drivers are missing and again, Lion will check and refuse to install on hardware that is too new in general.