I am not familiar with TX weather; but it being (hot)
south, it may be a good idea to put it in ground,
if succesful, it will perform better.
How old is the tree? You may need to winter protect it
for a couple of more year. I also hear that being next
to the Gulf, the weather sometimes gets out of hand...

Oh, I forgot to mention; if you decide to put it
in ground...

First 'score' the root ball top-to bottom
(2-4 opposite sides) to cut any 'circling' roots.

And while you are at it, do a quick inspection
for the dreaded RKN pest! If infected, you will
see blobs on them roots - if so, discard the whole thing.

I've had it for 2 years... I don't know how old it was when I got it. It was maybe 3-4 feet tall when I got it.

I know most trees need to be planted in winter, that is not the case with figs?

Austin is not near the gulf, it's about 4 hours inland. Summer temps are usually 95-105, winter lows occasionally to upper 20s, 30s more likely.