As so often, Sheldon Brown had straightforward good advice on removing such a fixed cup:
https://www.sheldonbrown.com/tooltips/bbcups.html
From his link:
He cautions about approaches to avoid: (including use of the thin wrench with 2 flats shown above, I believe):
Fixed cups have wrench flats, so you might think a very large wrench
would be sufficient to remove the cup. Not so. The cups are screwed in
too tightly, particularly since the wrench flats are only about three
millimeters wide-too narrow for a wrench to get a good purchase on the
cup.
Fixed-Cup Tools
Very elaborate, expensive, and special tools are sold to bicycle shops
for fixed-cup removal and installation, but they are not really
necessary. A few cheap home remedies can do the job quite well.
Vise use, as suggested by @BeginnerBiker :
The first home remedy demands that you have a high-quality bench vise.
Simply grab the fixed-cup wrench flats in the vise jaws, making sure
that you have good contact with all of the wrench flats' meager width.
Grasp the bike frame and unscrew it from the cup. This technique works
quite well.
Otherwise, the approach I used:
If you don't have a bench vise, or if the vise technique happens to
not work well on your bike, you can make a universal fixed cup tool
for less than a dollar (assuming you have access to a couple of large
wrenches).
The tool is an imitation of a $40 shop tool that fits into the fixed
cup as a crank spindle would, and pinches the cup. You apply
unscrewing force to the tool, and it transfers this force to the cup
with this tight friction fit.
The tool consists of a large bolt, a nut or two, and a few washers.
The size of the bolt is not particularly critical, as long as the bolt
is strong enough not to break, but small enough to fit through the
hole in the cup. I used to use an ordinary 1/2-inch, 13 TPI hex bolt
(also called a "cap screw"), which served me well for quite a while.
It finally met its match on friend's Schwinn that had an unusually
tight fixed cup; the bolt snapped in two before I could remove the
cup.
Now I use a 5/8-inch 18 TPI hex bolt 1 1/2inches long, with a nut, a
flat washer, and four lockwashers. The 5/8-inch size is the largest
standard size that will fit through the hole in the cup. This bolt and
nut both take a 15/16-inch wrench. With my 1/2 inch drive Craftsman
six-point socket set, the 15/16-inch socket is also the largest size
that will fit into a normal bottom bracket shell.
If you have some other brand of socket, check the fit before you buy
the bolt and nut-you might need the next size down (9/16-inch).
Bolt as bottom-bracket cup tool
Home-made tool shown in cutaway bottom bracket cup. Nut on the Outside
The bolt is put through the cup from the inside of the bottom bracket,
and the nut screws onto it from the outside. There should be washers
on both sides of the cup. You will need a socket wrench and a short
extension to tighten the bolt from inside the bottom bracket, and a
box wrench for the nut.
A large socket wrench will not fit into the cup, so you will need some
small-diameter washers to space the head of the bolt out so that the
socket wrench can reach it. This is what I use the four split
lockwashers for. I didn't need lockwashers as such, but the readily
available flat washers were too large in outside diameter to fit into
the cup.
To remove a right-threaded fixed cup (French or Italian), tighten the
bolt with the socket wrench from the inside of the bottom bracket,
holding the nut with the box wrench. Once the bolt is as tight as can
be, keep on tightening it, until the cup screws itself out. Once the
cup starts to move, turn both wrenches together.
To remove a left=threaded (British, Swiss or Raleigh) fixed cup, hold
the bolt with the socket wrench and turn the nut with the box wrench.
Use the same tool the opposite way to install the new fixed cup.
You might worry about damaging the cup, but this is not likely to
happen. The washer doesn't usually come into contact with the bearing
surface. Even if it did, bearing cups are made of extraordinarily
hard, heat-treated steel, much stronger than that used to make
washers.
PS - the bolt, nut and washers probably costs a few dollars these days.