Ignoring Self-Signed Certificates from Powershell Invoke-RestMethod doesn't work (it's changed again...)
I think I have narrowed this down to a change in the web service that I'm calling. Doh!
The Disable-SslVerification and Enable-SslVerification function that I listed in my question are still the best way to go and seem to work.
I look forward to the -SkipCertificateCheck switch mentioned by Bacon Bits in the comments. Then, we can stop hacking. =)
Hopefully this question is valuable for people who are trying to solve the same problem but run into the An unexpected error occurred on a send problem.