network gap exception "transferable" between insurance companies?

Hi insurance knowledgeable people on reddit!

So, I’m between stages of a two part surgery with a surgeon who’s in private practice and not in network with any insurance. The surgeries are done in a big medical center that’s in network with all major insurances, so OR, anesthesiologist, o/n hospital stays etc are all covered by insurance. My surgeon’s office got a network gap exception for his fees for my surgeries from my current insurance. I currently have an Aetna POS plan through work. Now, my boss is relocating to a different work place in 3 months and wants me to relocate as well, ideally before the end of the year. The new work, a medical center, offers BCBS POS plans and some internal plan with no costs, super low co-pay, but aside from few exceptions like ER visits it’s limited to doctors within this medical center.

With the short time frame of being asked to move in 3-4 months and my surgeon requiring at least 6 mo ths recovery between stages (im only 9 weeks post-OP now), I doubt I can get the 2nd surgery done before changing work places and thus insurance. What would be my best option here?

1.) I strongly doubt it, but is it possible to transfer a network gap exception? 2.) Would having had an exception with one insurance for that surgeon and procedure make it easier to get an exception with a new insurance for the follow-up surgery (established care, surgeon is best qualified to finish what he started,…) 3.) In general, would it be easier to get network gap exception from a big insurance like BCBS, or would the medical center internal one actually be easier (no one there provides that specific care). 4.) Should I look into keeping my current plan a bit longer, until I managed to get the 2nd surgery, using cobra? Is Cobra even an option if I loose coverage by being relocated to a different employer, even though the new employer offers insurance?