I'm not a big fishing enthusiast, but what precludes you from fishing from the banks of Lake Travis in Austin, or the banks of the Colorado River, or the banks of Inks Lake (a State Park north of Austin, in Burnet), or the ponds at Bastrop State Park in the Lost Pines (about 40 mins from Austin)...
These are just a few off the top of my head. We list a goodly number of
lakes and parks in the Austin area you can check out. I can't think of any except Hamilton Pool Preserve that *don't* allow fishing.