I have enjoyed my long distance with Thai from the UK. I have also flown EVA a couple times for short haul. I am ready to try them for long haul, I was impressed so much!
I think as long as you leave for more than a day or 2 (I would personally suggest a week), then you should be fine. You DO plan on getting extensions, right?
I personally would try a border run, not a bounce. Stay somewhere for a week. When you re-enter, make sure you have a departure flight. You will get more scrutiny for an almost expired passport. But they may let you in if they assume you are leaving within the 45 days.
Still trying to piece together how you got yourself in this mess.