re: LSU Purple

The fact it hasn’t grown an inch in over a year seems unusual. What time of the year did you plant it?

Have you planted it in area with sufficient sun - at least a 1/2 day sun, whether it be morning, mid-day or afternoon? Is the planting location low and holds water after it rains or is it well drained? If you suspect the area is not well drained, and perhaps the fig is planted to low, you can remove the fig, build up the area you have it planted perhaps 6 to 9 inches high (perhaps 3 feet in diameter) with “garden soil” and re-plant it in the raised area. You can do this anytime now since the fig is dormant, small and young in age.

Sometimes it takes trees/shrubs a year or so settle in and grow, but my experience with figs, and the LSU Purple in particular, is they grow pretty fast.

And I see, as others recommended, that you have verified it is living. The fig varieties recommended for Louisiana, including LSU Purple, are very cold tolerant.