Have it swing out. No need for double doors.

Replacing a door to an en-suite bath with double doors is a workable solution that I have done before. I wouldn’t recommend it for a hall bath or powder room, but I think it work for your purposes., assuming you have the clearance on both sides in the bedroom. You will purchase a pre hung double door and have it installed. Not a huge ordeal, they will take out your old door and replace with the double doors and replace the trim. A big box store like Home Depot could do it, or you could purchase it and hire your own carpenter.

I missed the sentence about having room for it to swing out. That is the best option.

I'd put the storage over the toilet, you don't have any room on the towel bar wall for anything meaningful. Swing the door out for added ease.

Just changed one from swinging in to swinging out. They had to rip out the door frame and put in a new one but that is because my apartment is vintage and has really specific, historic frames and hinges. I would think it would be easier for most homes.

Why would you want double doors???

Even if you change swing it doesn’t seem like there is much space for storage? You would be blocking entry. i’d just do over toilet storage.

That looks like a very high quality door and door hardware. Will the door hinges be on the same side of the door, and the door swing in the mirror image it swings now? The hardware does not look like it is necessarily reversible. (Like the curve of the lever.)

I have homes with small bathrooms. I wish i could swing out for one. We did a pocket door for an ensuite guest bath.

Unless a bathroom is a nice size, swinging out makes the “feel“ of the bathroom more spacious as you enter and close the door.

If you have dead space outside the bathroom perhaps a nice tall cabinet on the left before entering the bathroom? Or what is behind the wall that the door opens into, could a recessed cabinet be added instead?

Changing the swing on the door doesn't net you any additional storage space.

Better to take advantage of the wall space above the toilet for storage.

Mount a wall cabinet about the toilet. It will provide you with a good deal of storage.

Somehow double doors don’t feel private enough for a bathroom. I agree with Beverly’s suggestion.

You do not have space for a storage of any kind where the towel bar is it will just block the entry out swing or not .

Is this a primary or guest bath? Either way, I don’t see any way to add storage unless it’s above the toilet. Bathroom essentials can be stored in the vanity. Multiple towel bars for the obvious.