Maybe not. Actually, I think you might have made a good choice for goats, as long as the fence has a buffer around it outside of which you could manage any escapees. It seems to me from here that if you plan for containing the forage you may be successful in providing higher production for your goats without causing a problem. I imagine you do have a fence because I would think that predation of your livestock would otherwise be a problem.
Most native grasses would probably be insufficiently productive for goats with the possible exception of California Brome (Bromus carinatus), which is a pretty good forage grass and a perrennial, thus it would not require reseeding every year (although it only lives four or five years depending upon how you treat it). The seed ain't cheap, but it's getting better. OTOH, although the seed you bought is probably cheaper, if your goat over-grazes it, you would have to reseed every year.
In any case, your success in managing forage for your animals would depend heavily upon how you manage the way they graze (eg rotational grazing). I don't know how much time you intend to spend doing that, but a lot can be accomplished relatively inexpensively with portable electric fencing.
We have a great 8 foot predator fence around the orchard, but haven't gotten into the portable fencing yet. That's next, because I want them to graze on the poison oak, berries, etc. around the property next to the road. Tying them out there just doesn't work. They always manage to nearly commit suicide with all the twisting around they do. (these are not very smart goats)