OK, let me explain: yes, it's considered Int'l if you are talking going outside the U.S... as well as ICAO standards, etc., it's all int'l....
...however, with regards to bid status, pay, etc.... it's considered Domestic. i.e. we don't get paid the $4/hr override to go to Mexico or Canada. Mexico/Canada layovers show up on the same bidsheet as Tucson and El Paso. Jepp revisions include US/Mexico/Canada revisions.You don't need an Int'l checkout to go to these places, and you don't sit through the 4 hour Int'l course either. It's all part of the Domestic system.
Anything OTHER than US/Mexico/Canada will only show up on the Int'l bid sheet, and if you are a Domestic pilot you cannot bid anywhere outside US/Mexico/Canada.
I hope that clears it up.
To answer MOST of your original question, I don't believe that there's a "regional" airline out there that has what you speak of. In the traditional sense anyway.
Sure, AE may get a higher perdiem for going to Mexico/Canada, whatever. But I'd be hard pressed to find a "regional" that ACTUALLY has separate bid categories/status, as well as route/airport qual to go to these "non international" international airports you speak of. YES, I KNOW that AE has their Caribbean stuff. YES, I also KNOW that ASE is a special qual airport for the airlines that do/have operated there. I also believe that Horizon has a special qual field(S) in Canada. But see below;
For the most part, all the sh1t is the same, you DON'T get paid anymore, nor get blocked/locked into bidding certain "categories/statuses.
At my company, ALL pilots go though the int'l training section. ALL pilots get bombarded with the same jepps causing a stupidly heavy flight case. ALL pilots are expected to fly ANYWHERE in the system we go to without ANY additional override/bid status, or whatever else you can think of to separate pilots going into whatever airport.
However, I'd be curious to see which "regionals", if any, have what you ask of. IF they exist. But ONLY in the way in which you reference.
Remember, flying 4 hours, clearing customs multiple times a day, as well as hitting 3 countries in one day is just "regional" flying.