In would email back and point to the phrasing on the website that says that the fight must be operated by a OW carrier or an affiliate, and point out that Republic is a regional affiliate of AA.

You can also try calling.

In the past I've had really good results with AS customer service, but from recent forum posts it sounds like the front line staff have gone downhill. Probably a lot of new hires with insufficient training. Just keep trying and escalating.