I posted a while ago that I used my 10mm AR in a steel shoot and it was subsequently banned for being too powerful. At the last steel shoot I noticed someone using a .41mag in the revolver class. After inquiring, as magnums and 10mms are banned, I was told he was using downloaded ammo. I was also told I could do the same thing with my 10mm AR.

What I'm considering is loading up some 180gr bullets with a fast powder, like BullsEye, and seeing if the gun will run. My thinking is I can get enough pressure to cycle the action without the speed. Basically, I want to keep it in .40S&W carbine range.

Anyone have any thoughts or see any problems with this?

Not at all. I personally wouldn't use bullseye due to how little is used. With so many powder options you could use many other powders and use up some case capacity.  But that's just my thoughts.

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Did you ask about the Power Factor that would be allowed?  If they have a particular PF that would also help you determine the velocity to which you will be able to work with...
Well Bullseye may be fast but there may not be enough residual pressure to cycle fully. 
You will likely need to experiment some using 40S&W loads to see.
Don't know what powders you have on hand but you will want a good clean burning powder that produces enough gas to cycle the firearm.
Powers like Power Pistol (6.4 grains), BE-86 (6.0 grains), HS-6 (6.6 grains), AutoComp (5.8 grains), CFE Pistol (5.6 grains), 244 (4.6 grains), Silhouette (5.6 grains) could be the ticket to what you are looking for...

Thanks for both replies. Was thinking BullsEye because it actually burns pretty cleanly at higher pressures. I get the case capacity issue.

No chrony at this particular club, so no power factor measurement, just the honor system. Others that do chronograph rounds won't let me shoot the 10mm carbine at all.

I have Power Pistol, BE-86, and Silhouette on hand. It's a little weird for me because I've never taken this direction, downloading 10mm, before. I've done it with other calibers, for various reasons. I'll guess I'll just have to load some up and see if they run the gun and what they do over my chronograph.

I probably should mention my carbine has a 16" barrel, so there is likely some coasting going on, even with full power 10mm loads. That was part of my thinking with the BullsEye. Maybe enough pressure to cycle the gun and have the bullets actually slow down in the barrel, maybe, kinda, might work???

The only way to know is to work down the loads in your gun. Except only under a very narrow range of conditions would a bullet be slowing down in the barrel. I might be inclined to first try an inefficient medium burner (more gas) and working it down.

Nothing wrong with trying Bullseye and seeing how it goes. I've loaded thousands of 10mm loads with quick burners for gamer loads. Not in a carbine though.

 a down powered load is still going to be moving quite quickly..
   My 16" 357 lever is at 1750fps while my 6" revolver is 1300fps
with the same factory ammo
  I know its not going to be the same as your situation.
 
    My buddy has a 10mm carbine and now alot of the plates are cracked.

When I want a lighter 10mm load, I use the same 180 grain bullet, but switch to AA#5 powder.  But semis don't have the luxury of light loads like revolvers do, so you may not be able to do what you want.

Have you looked at frangible bullets or powder coated bullets so as not to have to download too far.  Curious how this turned out

The precursor to the .40 S&W was the 10mm FBI load, a 180gr jhp at 950fps.  I'd try a 180gr fmj over 6.0-6.5gr of Unique and see if it runs your carbine. 

Thanks for the suggestions, but the Covid thing put this on the back burner for me. All shoots have been cancelled. FWIW, my 10mm PCC is gas operated and will not cycle without running near max loads.