I've managed to get this working, even though I'm not gonna use it anymore. I don't really have a dynamic IP, but a failover situation, in which it might swap between two different static IPs.
So, I'm using a previously existent, publicly trusted cert from my company. It has no IPs set as SAN (only a wildcard as DNS name), and it has client/server authentication in its EKU.
I've done so many things to make it work, that I might be forgetting something important, but I remember that importing the server cert into the "computer->personal" (don't ask me why) folder was key to make it work. Probably there's a better way of doing this. One thing though: I've been doing preliminary tests by switching the IP resolution directly in my hosts file. Didn't get to the point of using DDNS.