Can you dry out pickle brine to leave a seasoned salt?

Reasonable is subjective, but as to safe the answer is yes. The point of pickle brine is to create an environment that will kill foodborne illnesses, and as it gets more concentrated through evaporation it's just going to get stronger.

Things to consider:

1. it's going to take awhile for the liquid to evaporate. You can speed this up by putting it in a flat pan in order to maximize surface area. You could put it in a low oven, but before you do that read point 2
2. It's going to stink, and could irritate your eyes and mucous membranes. Your house may spell of vinegar for days afterwards, so you'd be best off doing this outside if possible
3. It's not going to save you money. Salt and vinegar are cheap enough that going through significant efforts to re-use them are a waste
4. It's not going to give you a product you can't easily and cheaply buy in most places. Think about salt and vinegar potato chips (crisps in some places). The flavoring is just salt and powdered vinegar (sodium acetate), which you can buy at low cost, or even make at home

So it's certainly do-able and will give you a safe end result although it may or may not be worth the trouble, that's up to your judgement. I'd certainly be curious, and if I had a warm, dry place I didn't mind smelling of vinegar for awhile I'd put it in a pan and see what happens. Remember, you are concentrating salt (an electrolyte) and vinegar (an acid) into something fairly corrosive, so I suggest using glass or a non-reactive material.

Why not? I like the creativity. "Pickle Brine Salt". You could try pouring the brine into a shallow container and letting the liquid evaporate. I am guessing the yield would be pretty low, so reasonable? Maybe not, as you could probably create the flavor profile more easily in other ways. Safe? I think your only worry would be that you could grow mold before evaporation. Maybe a dehydrator to speed things up?