This has always stumped me (along with wheel builds). I'd like to recover a Campy BB from a trashed frame. I've never been able to remove a fixed cup. The one in question is on the drive side and I've tried turning it clockwise. I have a cheaper wrench that came in a $100 bike kit. I cannot keep the wrench engaged while I apply force to the loosen the fixed cup.

Is there a particular trick and/or tool that will help me remove the fixed cup?

An Italian threaded bb on the drive side is tightened by turning it clockwise...for more leverage try using a longer wrench or an extension on your current wrench handle (cheater bar, piece of pipe, etc).

Sheldon Brown has instructions for building a DIY fixed cup removal tool:

https://www.sheldonbrown.com/tooltips/bbcups.html

I built a tool to his specifications (about $5 in parts at the hardware store), and it allowed me to exert lots more force than an adjustable wrench or even a bench vise. That said, I still wasn't able to remove the cup and was forced to let my LBS do the job for me.

I've had a J.A.Stein fixed cup wrench clamp for years. I love that thing.

A bolt and a stack of washers could accomplish the same as a J.A. Stein wrench clamp. The washers would hold the wrench (crescent or BB) in place while you apply torque.

Just to repeat the obvious. If the markings on the Campy fixed cup are 1.370 x 24 tpi or 35 x 1G then it is a left handed thread and turning it clockwise (from the driving side) should loosen it. If the markings are 35 x 1 or 36 x 24F then it is a right handed thread and turning it counter clockwise (from the driving side) should loosen it.

in the shop we use that Stien cup tool but our wrench is bolted to a 2 ft cheater.

I tried to do the same in the past and failed. It's very inconvenient and expensive to go to the LBS. The other mechanic (defunct LBS) is a nice guy and economical - but hard to pin down.
I've emailed directly to purchase the tool. I'm also buying a closed end BB cup wrench to replace the piece of junk I own.

Thanks - its 1.37 x 24

Sure:

Going to an enclosed wrench and using the JA Stein tool should do the trick but if it's still stubborn, I've had good luck adding a 3lb dead blow hammer and/or a propane torch to the mix.

i've gotten really good results using a homemade headset press tool to help remove super stuck fixed bottom bracket cups. its the one made of 18" of threaded rod with two big washers and nuts (two plumbers reducing couplers optional) costs about 5.00 in parts at the hardware store and works beautifully.

get everything pulled off the frame except the fixed BB cup. spray some PB Blaster / liquid wrench if it looks corroded. keep one nut and washer on the threaded rod on the side where the adjustable cup used to be. the big washer should be up against the bare frame BB with the nut on the outside. push the open end of the threaded rod through the hole of the fixed BB cup and add the park tool fixed cup tool that fits and then add the other big washer and nut. tighten it down slightly past finger tight enough to hold the tool on the fixed cup so there's no potential to slip. get a good angle on the tool so you can put your body weight on it. you can also alternately sit on the floor holding the frame and push with your foot on the tool using your leg strength.

i've rebuilt about a dozen 25 - 30 years old bikes and it hasn't failed me yet.

I always had good luck with a big adjustable wrench to remove even the tightest BB fixed cup, type with the two flats on it. Trick is preparation and patience.
It is always tempting to attack the fixed cup with a tool like an adjustable wtrench like an ape, but the best thing to do is to take your time and first stabilize the frame of the bike by laying it down on it's side on the ground, and making sure that it will not see-saw or move around on you when you apply force on it. I always just bunch up a big old blanket or quilt on a carpetted floor and lay the frame on it. Careful adjustment and tightening of the large adjustable wrench (Ideally the size with a handle over a foot long) will have the same results as doing it with a proprietary fixed cup wrench. Just do it slowly and easy with constant checking on the adjustment of the wrench so you avoid having it pop off on you (also push down on the head of the wrench against the BB face while you are applying torque. It's when the wrenches pop off on you and you ruin the flats on the fixed cup that the real trouble starts.
BTW, it will be good if you also pre-treat the fixed cup by spraying some penetrant lubricant on it from inside the BB shell. Liquid Wrench, PB blaster or Freeze-Off should work nicely.

Good luck, and remember to take your time and it should work out.
Haven't had a fixed cup beat me yet in over 20 years of wrenching on bikes.

Chombi