Removing square taper bottom bracket cartridge after stripping notches
I had something like this on an old folding bike.  The owner said that he had it installed by a gorilla using loctite threadlocker.

So the fix was to use leverage, but to control it so the tool did not slip.  We had the non-drive side off and removed the axle, and reinstalled the shell temporarily.  There was some threadded rod through the BB, and we used a stack of big washers to spread the load.  This held the BB tool (a park HCW-4).   Then we used a pikey hammer to gently turn the cup.

Sounds like you have the cranks off but neither side cup.  So your best option is to rig up something using the BB itself as the shaft that holds the tool in place.  Some washers, or a short buttpipe/bushing might be all it takes.



As promised, here's the tool I made for taking off a cup that is stuck.

Here's the normal bottom cup tool with two flats on it.


Completed tool ready to turn:


Photo of it without the middle tube in place:


Making the two washers - started with large 2" washers from the junk drawer.  One needed an 8mm hole in the middle for the crank bolt, and the other need a 26mm hole to clear the BB axle.  Somewhat rough way to do it, but it works.


The purpose of this is to hold the correct too in place while you apply leverage.  Also means your fingers are nowhere near the pinchy bits.  When working alone you can now use hand leverage, and then a progressively increasing series of hammer taps to work the cup loose.

NOTE the drive side cup is left-hand threadded.  So turn it the opposite way

If you're talking about the cup on the non-drive side, then its often a different fitting, with a lock ring.  You can either do exactly as illustrated, but use a park HCW-11.    Or you can use a pair of stiltsons or some sliding lock-jaw pliers on the locking ring part.