Welcome to the forum.

The Desert Eagle in .357 Mag is incredibly easy to shoot.  It's a big, heavy gun with a gas op recoil system that makes shooting full-powered magnum rounds feel like shooting a .22 mag pistol.  And while the .44 has a fair bit more recoil and muzzle flip, I can shoot loads through the DE all day that give me a sore hand in a revolver or single-shot after a dozen rounds.  

I would be concerned about the plating you're finding that's going through the gas system.  I have one barrel that the gas port seems to shave the bullet as it pushes past it.  This was the case with both copper plated and jacketed bullets.  It cycled fine at first, but then started getting less and less reliable.  I'm guessing some of those copper chunks were getting hung up in the gas tube.  I eventually sent it back to MRI. They reported to have changed the recoil springs but nothing else.  And it worked better after that, but I was still getting some copper chunks coming through the system.  I ended up buffing it with a rubber polishing tip to try and take the sharp edge off the gas port.  It seem to do better, but I want to put a few hundred rounds through it before I'm convinced.  

In other words, keep an eye on the chunks going through your gas system.  Copper electroplating is some pretty tough stuff.  I don't know that the powder alone would do it.  It could be that the other powders are just doing a better job of blowing it out of the channel under the barrel.

As for powder selection, I'm not surprised HP38 and Unique didn't work well for you.  The burn rate is too fast. They don't produce enough gas volume to reliably cycle the action.  The others you listed are all good choices, and I would add H110/Win296.  L'il Gun is one other powder that I've used in .44 Mag that shows some promise, and Viht N110 works well also.

Powders that I've tried that didn't work so well were Blue Dot, HS-7, N350, and oddly enough, AA1680.  

One thing of interest, the Accurate Arms Loading Guide No. 2 (© 2000) has sections specific to Desert Eagle loads in .357 and .44 Mag.  One assumes .50AE is all DE-appropriate loads.  They aren't anything special that you can't find elsewhere.  They just listed those with bullet weights and powder burn rates appropriate for the big gas-operated pistol.  I ended up finding a copy on ebay, but as noted it doesn't have anything you can't find in a more recent book.  

Well done on finding the 10" barrels.  They have been getting difficult to find recently and command a premium when they do come up for sale.

And what "list" is there for 10" .429 barrels?  I had read somewhere that MRI was thinking about making a run of 10" barrels, but I haven't seen anything on their website.