Hamilton Pool Road has recently been remodeled between RR 12 and Hwy 71. It has a center turn lane and shoulders wide enough in most places to allow you to ride with 3' clearance to vehicles in the traffic lane. It carries a lot of traffic so it won't be a quiet experience, but you will not be sharing pavement with vehicles.
At the west end of Hamilton Pool Road two possibilites exist to connect to Fitzhugh Road :
1) RR 12 with a wide shoulder.
2) Bell Springs Rd & S Oak Forest Rd : no shoulder, traffic is primarily local acreage residents and speed limits are 45 mph or less.
At the east of Hamilton Pool Road there are two possibilities to connect with Bee Caves Road quite a few miles west of the Southern Tier intersection of Bee Caves Road and Barton Creek Blvd
1) Hwy 71 between the western end of Bee Caves Road and the eastern end of Hamilton Pool Road : no shoulder where 3 lanes in each direction, shoulder not much wider than handlebars where 2 lanes in each direction, multiple lanes of heavy traffic, multiple high volume cross roads with speeds at 50 mph or greater
2) Bee Cave Parkway : Eastern section share one of 2 lanes of traffic in each direction, traffic volumes extremely variable by time of day & day of week, can be nearly empty or can be high enough that overtaking vehicles have to slow to your speed, signed "bikes may use full lane". Western seciton has shoulder not much wider than handlebars