Ben:
Austin's a great city, but I cannot imagine a reason under the sun or in the clouds why you would make AUS your day-trip destination, unless you just want to show off that you can make a turn-off from 17L-35R to one of the two, next-to-each-other-we-cater-to-jets-mostly FBOs, and enjoy buying avgas for $5.25. If you have a specific eason for going to Austin (like I do for every Univ of Texas home football game), that's different, but for the city of Austin, AUS is the only game in town.
Now.
Great day-trips from TKI?
Shorter than you referenced, but Granbury (GDJ) is about an hour or less southwest; arrive during business hours, and they will loan you one of their fleet of retired police cars gratis or, if you are going to arrive outside FBO hours, call ahead and they'll leave the keys in one for you. Old Granbury is a very attractive town, with a lively local theatre and the sometimes-famed Nutt House restaurant has, blessedly, reopened.
You mentioned Fredericksburg, and that's a fine trip, though it'll take you more than an hour by 172. Arrive lunch time, and eat at the very nice Airport Diner associated with the also-famous hangar Hotel - call first to check hours. Some nice historical stuff there, too, including the Chester Nimitz museum (F'burg is his home town). Lts of shopping (sort of touristy, but still nice) in town. Great airport.
Stop at Lampasas (LZZ), and gaze in awe at the self-serve pump with the $2.78 100LL.
Do a TGL at Greenville Majors (GVT) and se what sort of high-end military hardware L3 is working on that week. Then, go south to Caddo Mills and watch the glider boys play.
Land at Aero Country, and find the yellow, white and blue Mouse, tell Bo, "Howdy." Of course, every time I land there I expect to see him, and have not yet. Might help if I called first...
So much more to do, but I am too tired to thnk of it all, so I pass the baton.
As for the WX, in late November, you can pretty much count on temperature between 15f and 85f, freezing level somewhere below 12,000' and above 10', and all couds will definitely be above ground level. Which is all to say: you just can't say. We are able to fly more than not, that is good, and while autumn / winter may have more no-fly days, the air is much smoother on the yes-fly days, so it works out.
Give a hint when you gonna show up...