Agreeing with Chrissy.
Make sure you have cash and generally a spendable financial cushion to ease future transitions. The job stress of managing a team and handling company politics is enough, plus location issues; keep financial stress down so your energy is up. Be open to growth, harvest the gains or at least the resume improvement, then move when the time is right for you. Careerwise it's easier to downshift than get promoted (again, politics!) so take the promotion.
Consider prepping some responses for instant use when key situations arise - maybe "I would love to complete the project and can run the team. Would I be compensated along with the responsibility?" (have a number or salary range ready in case they ask), for example. If they respond with blah-blah-blah-we'll talk later-you'll probably get a rasie after you finish the project, maybe "I'd like to focus on the job itself during this key development phase, rather than needing to explore offers when they arise. Who would be the decider on resolving a commensurate salary now?"
Extra idea: Until you are comfortable/experienced in runnning the team and sorting out the politics, read an Ask A Manager article every day. When needed, or when you have time to prepare, search for articles there related to topics of interest to you and your team. Every problem you face, Ask A Manager probably answered.
PS. Glad you're already looking. Sounds like you have the "common sense" (so uncommon!) and technical skill to succeed anywhere, keep up the good work.