I have some friends who were working on trying to open up LST to bikes, and what they found was that it was so poorly maintained that biking it would be a real problem. They figured that opening it to bikes would improve maintenance (by riders) but that was met with heavy resistance from hikers.
Same guys do a ride that includes both Double Lake and HSP trails, and small roads in between. I'll try to find details.
We used to do the same between Rocky Hill Ranch in Smithville and Bluff Creek Ranch in Warda and call it the Gutbuster. Years later, they modified the route, made it a race and called it the Excrucuating Exam or Torture Test. Both of those options are only about a half a day, though.
Another possiblilty is to pick up a Texas Backroads Atlas and look around. There are tons of backroads that you could probably link up together between other state parks without dealing with many cars or even pavement.
If your family are decent riders, you would have a blast up in Arkansas by linking the Womble and Ouachita Trails. Great camping also and definitely worth the drive up there.