I've been through this from a Mexican's perspective.

I basically have one passport with my birth name which I use for Global Entry, I got another passport with another name that I use sometimes to fly out of the US and I just renewed GE.

My previous GE card had a simplified version of my birth name that didn't match my GE passport or my European passport, but I always got TSA Pre (as long as the boarding pass was printed in the US)

I just took a look at my latest GE card and guess what? It has a name that doesn't match the previous card, or either of my passports haha. But since I had no issues in the past I think it will be fine, in my experience the most important part to get TSA Pre is to have the US-based airline print the boarding pass, if I get the connecting flight passes printed overseas I never get TSA Pre.

My workaround has been to download the US-airline mobile app and get the boarding pass in there, which always has TSA Pre enabled.