Is it possible to get a decent signal w/ high-level inputs and the Bose HU?

I have a fairly good signal to my sub with that setup, but subs are less picky about signal quality (unless gain and/or volume is maxed out). I want to swap out the front (and maybe rear) speakers. Can I keep the Bose HU and still get a decent signal w/ the speaker-level inputs on my amp?

I would rather put that money into better speakers, but I don't really know which way to go. Any opinions?