Are there any country bands with 'melodic' drumming, where fills are to be expected? Something ala Peart from Rush, where a big part of the overall sound is the 'melodic' fills - can't think of this in country or country/rock, bluegrass, etc.

The closest I can think of are not mainstream country bands, but more of an alternative rock/country hybrid - like The Blue Merle
Here's one where the drummer does very tasteful fills:
https://www.youtube.com/watch?v=YoWA_Lr6aoQ

Note that most of the fills are very simple - only one that I can count that uses rack toms.

I think you have to go to prog-country like String Cheese Incident, to hear fills aplenty.