I was under the impression that I did not need to register with the commune authorities if I have a carte de legitimation, which was also confirmed by my org's HR. Can anyone confirm either way?

And, if it turns out that I do need to register, which should normally be done within the first couple weeks I believe (I've been here for almost a month on a tourist visa [non-EU]), what will happen if I try to do so now?

Thats correct, you do not need to register if you will receive a carte de legtimation via your organisation.

Great, thanks!