If it were me, I think I'd prefer to fly in & out of the US, wrap up your things in the UK, and then return to the US fully. I don't think I'd take my chances with the additional processing & new medical! It takes a few weeks to receive your SSN & a few months to get your physical green card anyway.
That said, I'm sure your preference will be to leave the job sooner. Perhaps you can use some vacation days.