How feasible is it to go back to BS to get a second degree?
The main thing is to reverse your attitude a little compared to lower-level study.
You are not applying. You are considering employing these people as your paid servants to teach you what you want to learn. You will be paying them in money and in time.
From their side, their desire is to serve their students properly: that is, to give them a good education. They cannot do this if you do not have the background knowledge and skills you need to be able to understand the course. If you are the one sitting at the back of the room counting on his fingers while everyone else is polishing up on Fubini's theorem, it is a waste of your time and therefore of theirs.
Conventionally, their suitability to teach you is measured by the bachelor's course you have taken and the certificate you have stamped on your forehead. From what you have said, you are not conventional. That is fine. Most institutions recognise to some extent that &quot;entry requirements&quot; are convenient measures of suitability but not necessarily the only measures. If an institution is unintelligent enough not to be able to understand that, then it is a clockwork machine, not an institution.
You need to contact the institution in question (or each institution, if more than one). You need to find the right person to talk to to have an exploratory discussion. This is not - and should not be presented as - a clever way of circumventing the requirements.
To give a concrete example: I know someone who failed to get into Oxford to read English and gave up the idea of academic study. Some years later she realised that if she didn't use her brain, it would explode; and that History was her chosen subject. She contacted the best college in Oxford, explained the situation, went and had an exploratory meeting with the tutors, and they said (a) from motivation and intelligence she sounded a perfect fit but (b) to make sure her essay-writing skills had not been forgotten they would require her to take the entrance exam.
Find who the right person is to discuss with. Ringing the secretariat and asking is a good a way as any: they love being asked for advice. When you have your meeting, simply put the case as clearly as you have here.

Is there anything I can do short of going back and being a first-year undergraduate again?
If starting at the masters level requires more skills than I have, then (a) do you have recommendations as to how I can acquire those skills and (b) do you have recommendations as to how I can reassure you that I have acquired those skills?

Adult-to-adult you should have a much more productive conversation than if you approach it as suppliant-to-gatekeeper.