That is very similar to the story I got when I ran a running store. We were the only "true" running store in town, but there was another sports store that carried a lot of running shoes about four miles away from us. They carried tons of Asics stuff in baseball and the other sports, that the asics rep did not want to jepordize that account by opening another one. The asics rep was not a running speciality guy, just a general asics guy.
We were successful in transferring lots of Asics wearers to Mizuno, Brooks, etc with our knowledge.
In the end we would have loved to have Asics, but it is not a deal breaker if you have a knowedgable staff. Of course, you are always going to have those people that only want Asics and will not open their mind otherwise, but what can you do about it. You can only sell what you have.
I think that Asics has a huge account with Dicks Sporting Goods. Every time I go in there and ask about a shoe, the first thing they point me to is an Asics.
Good luck.