We had the same open bathroom problem with our former house. We hired a crew to reframe the opening and install a door. It was worth it.

They could also remove the arch detail and soffit. This kind of stuff was done in the '80's and '90's.

I don't understand your assumption that the door way can't be framed to hold a traditional door? The semi circle can be filled in. What's that bulk head for? Hopefully it can be removed and the wall will be flat.

A regular door won't fit right with the bulkhead/soffit. And it should swing into the bathroom but it swings into the space next to a bath so it's awkward. We would like to remove the soffit not sure what's under their or if we have the time/money to invest right now.

For sure fill in the arch

pocket door. you should be able to find the correct size for that opening. you'll need to remove some of the drywall on the right side, install the hardware into the wall, patch and paint.

we did one for our entry

I think I would sort out whether you do actually have the budget to remove the soffit first, because if a regular door "won't fit right" with the soffit (which I don't know what this means, is the opening lower than a regular door? ) a regular pocket door or barn door hardware won't fit either. And you will just be spinning your wheels spending money for something you are going to change later. A pocket door is not an easy solution, particularly if there is any electrical, ducting or plumbing in the way. And there is usually electrical in the way because of lightswitches at doorways. If you need a quick solution block off the archway and put a room darkening curtain over the door opening.