Where are you starting from? I don't have any specific backroad suggestions, but it looks like you could either take a "northern" route, staying north of the lakes a la 1431, or a "southern" route, staying south of the lakes, a la 71. It would be an interesting ride...a lot climbing.
Leaving from Central Austin, Lamar@Koenig, headed toward Deer Haven, just past Horseshoe Bay on 71. Tryng to avoid 71 as much as possible!