Venting a portable a/c into another room that is closed off with the window open
In theory this could work.  Probably not, and it depends a lot on the details. The biggest obstacle IMO is you need to cut a hole in your bathroom door to find out.
Let's solve the easy part first: You can repair the hole by camouflaging it.  Mount a door mirror over it, or cover it with wood and put a picture frame around the wood, like the apartment door in Friends.  Maybe something toilet themed.  Be creative.

Now for the A/C: An A/C moves heat and moisture from one place to another.  Usually to the outside.  The two most common examples of moving the heat to another inside space are 1) When they put a cashier's booth in a large underground parking lot and install a window A/C in it, venting to the parking lot.  2) Your refrigerator, which moves heat into your kitchen.
Here's the catch: In both of these examples the heated space is much much larger than the cooled space.  In your case, your bathroom is much smaller than your bedroom, and if you sometimes open your bedroom door to enter or leave it makes the challenge even harder by admitting new warm moist air.  It all comes down to how effectively that open window works.   If you seal both doors tightly and put a fan in the window, there will be no air flow.  The fan will mostly create turbulence.  It will help a bit.  But whether it will work enough comes down to experimentation.  If not, the bathroom will become so hot that the A/C will cease to work efficiently or at all.
An idea
Well, the best idea of course would be to use it the way it's intended.  But anyway.  Rather than seal both bathroom doors, you could open the window, install a powerful exhaust fan in it, well sealed all around the edges, seal the bedroom door, leave the other bathroom door open, and open a couple of downstairs windows.   This will create excellent air flow through the bathroom, and I think the A/C should work perfectly well.
Really?
The spectacle of a powerful exhaust fan installed in your bathroom window with sealant and tape, an extension cord running through the bathroom, the need to actively manage the entire system by keeping the other door and downstairs windows open, .... is all that really more aesthetically pleasing to you than just using the provided hose in your bedroom window?
My advice would be, if the entire plan is the one described in your Question, don't do it.  It's a big gamble to destroy your bathroom door for a plan that is unlikely to work very well, or at all.