How to navigate switching teams when working on a project that's dragging on?

I'm an E5 mobile engineer at a Big Tech company. Due to lots of manager attrition, I currently report to a hands-off Director with too many reports to have regular 1:1s. I found an awesome EM who agreed to let me join his team and promised me E6-scope projects on his team. My Director is his skip-level, so I'm staying in the same org.

However, before I could make the official team switch, my TPM loaned me to another team lacking mobile resources to meet the TPM's own OKR. He did not bother talking to the awesome EM or me beforehand. My scope on the TPM's project is E5 at most. Now that project is dragging on. It's already code complete, but they want to keep me on that project until it's fully rolled out. We're waiting for mobile adoption to reach a certain threshold before we can do a force upgrade. Due to the code chill around the upcoming holidays, we likely can't do the force upgrade until next year. In the meantime, the project's EM is asking me to investigate pre-existing bugs in their feature. The awesome EM met with the TPM and that project's EM to fast-track my transfer, explaining that he needs me for Q1 planning & our team's own OKRs, but the latter two insisted that I need to support their project until it's completely done, which includes the force upgrade. Am I stuck on this project until January next year or is there a way to switch teams more quickly?