I hate being cliche, but a chain is as strong as it's weakest link. If someone is slacking off and another person has to move forward in spite of that, I don't see the person moving forward being at fault. The one slacking is to blame. I would have no guilt or ill feeling whatsoever moving forward. You're not at fault or to blame. Seems like you're quite clear on what you want. She's getting in your way. Not the other way around.
But that wouldn't even matter to me. Playing in multiple bands is something I've done since I was 15 years old (that's 40 yrs. for anyone counting). I've played in as many as 5 bands at a time. I'm currently in 4, and if a good opportunity came up, I'd still be open to it. While I understand your uncomfortability about moving forward with others, I really feel those uncomfortable feeling are something you'd best accept and not allow that to get in the way of your actions. Playing with other bands isn't a sin. It's not cheating. And there's no reason you have to tell them, especially before anything even becomes of it. If it does work out with the other band, it's a good idea to give them a heads up... and if conflicts start arising, too bad. You want to gig, you should gig. First band isn't gigging, oh hwell.
As far as juggling multiple bands I usually stick by the calendar rule, whichever gig is booked first is the one I play... or I'm clear with the lesser priority group that I can't fully commit. Surprisingly, this works out more times than I used to think it would. Often times the bands I jump into have a bass player already (that I'm replacing) that can't commit to all their gigs, so they can easily sub if a conflict comes up. So long as the band stays on good terms with them, which I urge them to do

. The one thing that can get tough with multiple bands is rehearsal time. The bands I'm in all only rehearse before shows. In your situation, I'd be crystal clear that if the singer wasn't going to be there, neither would I, and that I'd be real pissed if she cancelled last minute. And I'd have to come up with a schedule that I, not they, were happy with. That's different for everyone, so I'll leave it at that.