The Domain is part of the North Burnet/Gateway development plan, and as such is governed by a plan that mandates and requires bike and pedestrian access. So far this has been all lip service and no action.
Three years ago WXNA proposed to The Domain and the COA that an abandoned underpass under Mopac that parallels the train tracks be turned into a bike/ped access point to the Domain. The COA bike folks embraced the idea, but thus far The Domain has not been forthcoming to develop the pitifully short distance between its fence and the parking lot or to at least remove the fence.
If the management of the Domain can be motivated to do its very small part, and the COA help out with some signage and minor improvements, this access point can be a done deal in months.
Here is a summary of the proposed path (it has a googledoc redirect and loads slow because of images, but lays out the path pretty plainly):