I think you have a pretty good plan. The only thing left out is income, but assuming that's good, with your already high limits and long history I think there's a good chance you could score some 15-20k+ limits with those CUs. A couple things I can think of:
1) You've got about 61k personal debt, and if you were to distribute that to other cards at ~50% util per card, that'd require about 120k in CLs. You already have 24k available on the Slate, so ~96k remaining. Given that, if you want to go all-in on the CC BT technique to reduce your debt, I think you should be more aggressive on new CU CC apps - maybe app for 5, hope for 20k each. This is not a common suggestion and I don't think everyone will agree with this (whether in principal or based on "economic conditions"), but I suggest it because it seems your profile supports it.
The reason I mention 50% util is because once you go much over that, in my experience your prospects for getting new credit decline. You should, of course, use as much of your CLs as you can to save interest, but once you go there, you're committed to working with the CLs you have until those balances come down. That's another reason why I think it makes sense to go "all in" with the apps now, because you'll likely end up BTing into some high-util balances, and as such you'll want to cool off for a year as you pay those balances down.
Some anecdotes: Alliant CU gave me 20k when I had inappropriate levels of util - I provided plenty of documentation, and they came through (they also send me regular 12mo 0% APR 2% fee BT checks). There are other big CUs: State Department, Affinity, and almost certainly some local ones in your specific geographic area. One of my state's employees CUs told me over the phone, "Well, you've been approved for the entire 50k credit line, how much would you like?" when I applied for their CC with 10% < aggregate util < 30%.
2) I'd try to get at least one card with a permanent 0% BT fee and low APR. Like a 6.99-7.49 range card, if possible. That way, when you want to shuffle these BT balances around towards the end of their promotional periods, you'll have a no-fee and low-interest way to do that. The other example would be moving between your CSP and Slate, for example, if that ever made sense. You temporarily BT it over to the no-fee card. One or two weeks of that low APR is a fraction of a percent in interest, much less than any BT fee you might pay. PenFed Promise is a great card for that, as they have an ongoing 4.9% APR / 0% BT fee offer, and there are others.
3) It's worth at least checking for a personal loan qualification. If you could score something in the sub-10% APR that'd be a pretty big win and would be a lot less complicated.
