Hello Cactus Jungle!
I recently moved my dinteranthus from a bright shade location to a full sun (bright shade for half the day) spot about two days ago. Today I noticed a strange lesion developing on one of the leaves (see photos). Overall the plant feels soft/a little mushy compared to before the move.

Any thoughts on how to approach this?

That’s definitely a sunburn. These are very delicate plants, so the prognosis is not great. The best I can suggest is to get it out of the direct sun, and spray with hydrogen peroxide to help it heal. If it survives the week, you can also spray it with some Neem Oil to help prevent any fungus.