I believe all Trifecta tunes reside in a place in the computer other than the other tunes, including the HP stuff, so you can only have one Trifecta tune and would need another basic tune to switch back to.
I happen to have the GMPP as underlying tune but there is no reason you couldn't get an HP tune that does the same thing as the basic Trifecta, except that it installs over whatever basic tune you have now, then add the E47 Trifecta on top of that. Then when you switched off the Trifecta E47 tune you would default to the HP underlying tune.
That would do exactly what you want, wouldn't it?