Although none of the three Pietta Remington 44s I've tried will take the Lee bullet without modification, some say theirs will. The Pietta Colt Army is set up for round ball, both with a long rifling twist and a plunger that's shaped for a ball. Depending on the date of manufacture, the Pietta Remington may have a long twist, but if it's not much more than about a year old it may have the faster, 16" twist which will stablize anything you can fit in there of a half-way sane weight. I modified my Remingtons, Pietta Colt Army and Uberti Walker to accept the Lee 200 and the Accurate Molds 45-200S. They're also good to go with the Kaido 220 and 240 grain bullets, which have more nose taper than my 200S. If you're loading off gun, that won't matter, but loading on the gun is easier and simpler once you do the mods.
I don't know about the Uberti Dragoons, but my Walker definitely needed substantial modification.
You'll quickly find that the Remington design is much more suited to easy loading of conicals (and paper carts) than any of the Colts. That is, once the loading ports are opened up and such comparisons can therefore be made.
Since you've been casting for some time, does that mean you have a lube/sizer? I size to .449" for the Piettas and .450" for Ubertis, but .450" or .451" will work in both. In paper cartridges, .449" works in both, but will be too small for the Uberti when used naked. Don't make the mistake of thinking that a conical bullet has to be oversized like a round ball, so it shaves a ring of lead - there's just no sense in that.
Since at Accurate Molds, the tolerance of the major diameter is all positive (e.g. .451" + .003" /- .000") then ordering a major diameter of .451" will give you more than enough diameter to work with, UNLESS you figure that at some point in the near future you'll be reaming out your chambers to .453" or some such. His tolerance on heel diameter is negative.
I made a mistake when specifying the heel diameter in ordering the Accurate Molds designs. I ordered them too tight, so any little teeny ding, or having them wrapped in paper cartridges, makes the heel too tight for easy loading. If you're going to order a custom mold like that, get a heel diameter of .440" or .442, up to no more than .444". Some of my Accurate Molds bullets load fine in the larger chambered Uberti, and they're tolerable but fussy in the Piettas when naked, but wrapped in paper cartridges they require my "heeling" process for easy loading in both Pietta and Uberti.
If I were designing a cap and ball bullet from scratch, for the Italian repros I'd want a longish heel (at least a tenth inch) that tapers from .440" at the base, up to about .444", with a small lube groove and a wide meplat (to make it shorter overall).
The Lee 200 is more of a tapered design, rather than having a distinct heel, and that's just fine. They seem to always load easily, but as I say I size them down also, to .450" or even smaller.
Life at the range or in the field will be more pleasant if your bullet heels slip in easily, and the bullets seat tight without shaving lead.
Also a more tapered-nose bullet will generally require less modification of the gun for easy loading than a more blunt (more wadcutter-like) bullet. That creates something of a conflict, because a longer bullet takes up more powder room in the chamber. That would only matter if you're going for a full power hunting load though.
That should be enough to chew on for a while.