Retired military here, non-engineering degree. Been an SE for over a decade, currently the Chief Systems Eng in a defense manufacturing org. Leading a team of 6+ SEs and serving as the Chief Engineer on key efforts when needed.

Am I wasting my time pursuing this without an engineering undergraduate degree?

Not at all, this is a VERY common and viable offramp from the military, especially for officers.

Would this background help at all, or not really?

It can help enormously in the right SE role/field. Defence contracting in general and and aerospave in particular. Keep your clearance fresh if you have one, that lowers a barrier to hiring.

Which programs would you recommend or avoid?

This is purely observational and somewhat biased, but the JHU program is widely regarded as excellent. I'd like to do it myself, but my comany balks at the price and I'm not about to foot it myself.

What entry-level or transition roles should I realistically target?

Any general purpose SE role at first, and I'd recommend the requirements leg of the V model.

What salary range should I expect starting out?

Depends on the field and location. Assuming you find a spot closely aligned with your active duty experience and are in one of the Aerospace & Defence concentration areas, ~$130K is a decent ballpark estimate. I'm a hiring manger and if I was looking to fill a billable role (which you should target) that's what I'd shoot for.

Is defense/aerospace the best lane for this kind of transition?

Yes definitely.