I'm headed in this direction myself, but not very far down the road yet, still dialing in the audio side. Done a ton of stills, live, and I know where issues lie there. I also study a lot of other people's videos from gigs I attended myself...sooo...take this w a grain of salt cuz you probably know better than me where the gaps are in my experience.
First off...Sonar X2 Essential set me back $100, and I have yet to find anything in Producer that Essential is missing that I need for live capture, and if I ever do, Essential files plug right into Pro if need be.
Second, I've done everything your original post requests, with just X2 Essential and Premier Elements. Just once, but it was easy, even tho the Premier manual lies off in my future. Premier Elements set me back $100, bundled w Photoshop Elements.
Third...nobody is gonna be happy with you being...in the way...while the band and studio staff is trying to lay down tracks, in an expensive facility, that probably was designed more for audio than video work, and probably less so while overdubbing, a fiddley process at a time when project money is running low. That means some advance planning on your part, including but not limited to, your trailing cables, any nouse you or they make, unforseen problems on your end, blocking progress on the audio side, any emag interference created by, or even blameable on you, justly or not, and the idea of re doing any takes just for vid, while on the $tudio clock. Not insurmountable, with planning, and a good attitude on your part. Possibly helpful would be a strong awareness on your part as a Fly on the Wall, seen but not noticed, as to the ebb and flow of the studio process, before you go live. A pre production sitdown, with the the three principles, talent, video, and audio, should help define clear priorities in advance, so that near certain compromises, aren't surprised at any times of stress.
Finally...I did not see major problems with sync. Any time at all spent watching Youtube, VEVO included, ought to make you aware that few competing vids even considered sync...drummers wailing on the snare visually during tom solos, chix doing handsprings w mics contacting the floor...silently...etc.
Still and all...Entry level software like Elements still provided numerous options to get clips n sound lined up. In fact... I'm not convinced you need to worry about music audio at all. You are shooting in a facility designed and tuned especially for capturing and editing audio. You should have full access, after the fact, to tracks, stems, rough mixes, final mixes, and masters. Not sure you need to sweat re-inventing the wheel, especially in that environment...however...
Some of the best parts of the recording process, visually and audibly, go down between takes...with the red light off. Its very likely you and all you to capture that. On the plus side, how hifi do lounge and tracking room and control room antics need to be to make their point? You might be able to get away w camera mics alobe. Or step up to a good mike wired to the cam inputs. Bit better, add a handheld Zoom type recorder, with a stereo pair to the above. Know where your audio feeds are in advance so you can tap in on your discretion, and test them thoroughly to make sure jacking in doesn't put artifacts onto the master tape. Choose carefully WHEN you jack in...the ONE TIME you don't want a pop is the 387th take of the bridge, the ONLY one that captured the groove w no screw ups...except for YOURS.
If you aren't in a hurry, you could chill till a studio set up for simultaneaos audio and video is online and dialed in...I mean seriously...outside your car, you goona listen, when you can listen AND watch...in this century? Hang tuff...I'm on it, we oughtta have mostbof the bugs out and go live by the end of this decade...2025 at the latest...I swear!