Do a hook and eye that is close enough to the edge that you can unlock it with a butter knife in an emergency. We did this with our kids' bedroom pocket doors and it works great.

why would you put a lock on it? these should not be used for public (other than say your own master bathroom) usage. they offer no privacy, sound proofing, etc.

besides which, should be obvious that if it's slid closed, then someone is occupying it.

I mean, it's already installed so probably not changing now. And even if it is in a master, if there are kids in the house that door being closed means absolutely zilch.

Not too late to change to a real door. Ever.

well just wondering where it is in the house. they don't make them because they were never meant to be locked.

you could probably make you own lock by getting those old style door locks. the kind with the hook and eye, or the sliding bolt type of lock.

Look at Artisan Hardware.

This blog used their Teardrop Latch on tneir BR barn door

But a Google search for lock for bathroom barn door should give you some options.

How did a door get spec'd, bought, and installed without this having been part of the plan?

And barn doors for bathrooms--bad idea.

I think a hook and eye is your “best” option here if you are going to keep the barn door. I am with @Sophie Wheeler in that barn doors are a terrible idea functionally. They look great on baths in showhomes and design blogs, but they are terrible for noise and light reduction and well, as seen cause issues if you actually want to lock it. A friend of mine bought a new build with a reclaimed wood barn door on master bath and soon switched out for a real door: middle of the night pee breaks or early risers meant a light flashing in rest of master or the rolling noise of a door or the noise of someone getting ready while other was still sleeping: more commonly all of those at once.