There are existing examples of 3-digit Interstate spur highways ending at roads that aren't an Interstate or US Highway.
I-575 North of Atlanta ends at a Georgia state highway.
I-385 in Greenville, SC ends at a regular city street with no route number.
I-140 outside Knoxville, TN ends at TN-33.
I-787 in Albany, NY ends at NY-7
I-790 in Utica, NY ends downtown at a jumble of state highways.
I-990 in Buffalo, NY ends at NY-263
I-184 in Boise, ID ends at S. 13th Street downtown.
I-105 in Eugene, OR ends at OR-99 downtown.
I-705 in Tacoma, WA ends downtown at a city street.
There may be other examples in the system. I just did a quick scan and found those.
I know about the existing I-12. Personally, I'd push for a western extension of that route to Austin, and failing that, I-16 and I-18 would be the next best choices (although this new I-16 wouldn't connect to the one in GA, and I-14 has already been proposed for a new route across the Deep South).
I don't see how the existing I-12 route could logically be extended to Austin without it running concurrent with I-10 for much of its route. For that matter, I-12 is barely justifiable as it is with having a 2-digit designation. If it were up to me I would have named the route I-410.