Well, current state of physics does say a kind of "warp" engine would in effect move a vessel 10X the speed of light, which would still mean a voyage of a few months from here to Alpha Centauri. The energy required is fairly vast, but at least within reach. Barely.Alright Im gonna borrow from BSG, and apply the idea to star trek. Make the show a hard sci fi version of star trek, with no aliens no ftl, transporters, force shields, artificial gravity, sub space commuication or other unknown techologies and stick to ones we know likely to exist.
Given that you're positing a solar system-based space opera (which strikes me as genuinely interesting), then the name "Star Trek" seems a misnomer.
I'll also note that Babylon 5 used spacecraft with rotating sections quite well.
But the "feel" of what you describe seems like something else. Something possibly very good indeed, but not Star Trek.
Part of me wonders why something more like the ability to cross in to other dimensions, and using them as bridges to cross vast distances. Not unlike Babylon 5's jumpgates.
I know that current research is involved in the concept of multiple dimensions, so that seems like a way to deal with the light speed barrier.
Though, I know it sounds a bit like hyperspace from various works.