Okay... no response after about a week. Good thing I'm far too self-motivated to rely on community support.

I went ahead, took the plunge, and followed my suggested script above. For the benefit of anyone else looking to try this tactic with a static IP setup,
don't bother. Just switch your IPs in your DNS and everywhere else you need to; it's probably less hassle.
For completeness, I can say that the process I outlined above works, but for
public IPv4 addresses only. It does
NOT work for public IPv6 addresses, or for private IPv4. (I don't rely on the link-local IPv6, so I didn't bother dealing with it.) Even though I swapped the private IPv4 and the public IPv6 addresses in all the necessary config files, the Linode Manager didn't seem to care; the "IP Swap" function only swaps public IPv4. So in the end, I still had to go in and change a dozen or so IPv6 related DNS entries, which only saved me about half the work of changing them all. While manually tweaking scripts and such to deal with changing private IPs wouldn't have been a problem, it would have been nice if the Manager had the capability to swap public IPv6 addresses as well. Maybe that's just "too new" of a feature for anyone to have considered it.

So bottom line: If you use static IPs and don't use IPv6, my outline above should work. If you use static IPs and
do use IPv6, don't bother with an IP swap and just change your DNS entries. You'll have to change your private IPv4 addresses in all your scripts regardless.