MaverickPrime,
I know nothing about visa issues but suspect a regional would be your best option. Your qualifications certainly appear to fit what they want and then some. All the regional pilot hiring websites have contact info for their recruiters/HR people. I'd suggest contacting every one of them and ask your questions. Don't be fussy...contact every one of them...even the bad ones. Be prepared to briefly explain the conditions/limitations of the H4 visa as they may not be familiar enough to see if your visa status might fit in their operation.
Good luck in your hunt.