You are wasting your time applying without the first class medical. At my airline all fleet types do some kind of international. Be it islands or overseas. That means that all pilots are type rated in their aircraft even as new hires so you can fill the role of PIC on IRO duties or whatever. That also means all have to keep a current 1st class. The old days were different but think all the majors require it now.

Once you are here we do have some older pilots that have something pop up that prevents a 1st class and they can continue to fly on domestic only stuff as F/O's but those are special cases.

Do whatever you need to, but get the first class. Otherwise in my opinion as somebody with 2 decades at a major you are just burning up your applications on a sure fire TBNT letter.

EDIT: Can you get the first class through the base doctor? I would think any US base would have somebody that is an FAA AME.