Lack of a period of winter dormancy is not great but certainly manageable. If your cutting is large enough, you can plant it out directly as soon as the weather/soil conditions allow, after sufficient hardening off. If not large enough (I wait until the root system fills a IG container - typically one full gowing season), then I would suggest potting up. But again, the vine needs to be outside so follow the same weather clues and hardening off process and tend to the container as if it was an actively growing, planted vine.

Yes, pot it up and then put outside at the proper planting time. I might let it grow a bit larger/bigger root system outside in the pot before planting it unless it turns into a pretty well-rooted transplant prior to spring planting time.

Congrats on rooting a cutting!

Now I just need to keep it chugging along for the next month or two until it can go out!

Those clematis do well in gentle Texas style winter dormancy where they never loose all their leaves. I would not worry about that. One could always put it in a cool spot now for the last of the winter/spring.