I believe the best way to gain access is to become known as a resource to the land managers, rather than another problem user. The best way to become known is to call the agency and volunteer to do trail work in the summer months, do educational talks on sled dogs and our sport, and participate in the access issues during planning meetings, etc.
Mushers are a pretty independent bunch, all in all.
My answer here in Maine has been to join both the sled dog and local snowmobile club to help protect access for my dogs through voicing my opinion. Good luck to you in your efforts.