Still hoping for some advice as we need to tile our niches soon. I have been unable to find any instructions or photos of bars installed inside a niche. Maybe this can't be done?

You could buy some chrome tubing, the kind used to feed water to faucets, and cut it to the length you need. Would the rod be supported solely by the thickness of the holes in the tile? I think it would be strong enough that way, it won't be holding much weight. I'm thinking you could mortar in one of the tiles with the hole, say on the left side. Put the rod into the hole, then a flange over it to cover the raw hole. Then slide the flange for the opposite side on, facing inward, then the other tile with hole. Mortar up the right side of the recess and carefully nudge it into position with the rod already installed. Does that make sense? I'm thinking you should cut the rod a tiny bit shorter than the untiled opening. So if the opening is 14 1/2" wide, and the tiles 1/4" thick, cut the rod to 14 3/8" long. You could experiment with dowels first to find the best length. You want it to be just shorter enough to be able to slide into position, but not so short that it can fall out of the holes.

If you don't want to use a flange, cut the hole nice and snug to the size of the rod and use silicone caulk around the rod once it's in place.

If you walk up and down the aisles of a Home Depot with an idea in your mind of what you're looking for, I bet you'll find a few options. Sometimes they appear in unexpected departments, so take a good look. But plumbing is probably your best bet.

You could also consider installing a simple style of towel bar against the back of the niche or the top even (hanging down).

Thank you Egganddart49 and Pricklypearcactus for your ideas. I will consider both of those options.