I must admit that I am really a double rifle novice, even though I own a few. I am a custom bullet maker and lifelong reloader for 45 years. It seems to me more than any other rifle type, doubles require a lot of "buy and try." Like others have said, I would not shoot ANY overbore mono-metal bullets, EVER, in any double. These fine guns are just to expensive to take the chance, and you could start a progression of ballistic consequences that could leave you chasing your own tail for a while.
1. For pure mono-metals, or premium bullets with mono-metal bases or mid sections, stick to .410,
2. For lead core, buy and try. You can always try them at .411 since your gunsmith says they are safe, then order a custom bullet sizing die to reduce them .0005, or .001 and try that. Stay with whichever regulates the best AND has best first shot accuracy.
As stated, I am a custom bullet maker so I swage my jacketed bullets to .416 for use in my .416 Ruger, Then run the bullets through a reducing die to get them to .411, .410, or whatever I want for my 400 Whelen. There is a much greater selection of .416 bullets if you have to buy them, and they will easily reduce to .410 or .411. This opens up all new possibilities, especially with supply chain issues being what they are. Even though I make my own, bullet swaging dies are costly and I have no problem with using reduced bullets, as long as you do not reduce a jacketed bullet more than .008 or so lower than original diameter. I only need one set of swaging dies for .416 to .410. If I have to reduce more than that, I buy a new set of swaging dies in the smaller caliber. If there is ever any question, I will order swaging dies .001 bigger and then reduce if needed. Reducing dies are cheap and time to reduce is negligible. Some calibers have a range of bore sizes depending on age and manufacture of the firearm, like the 450/400.