You'll need to slug your bore (google it, lots of info) to find out what your barrels groove diameter actually is. These can vary hugely, especially in older guns, so don't assume anything until you slug it. You then want a bullet that is .001" to .002" or sometimes .003" larger than the groove diameter you find with the slug. Also it's best to use a micrometer to measure the slug as many calipers have a reading error of +/- .002" from the get-go. I usually use 00 buckshot to slug bores of 8mm/.32 or less but for the 44 you can have a harder time finding a soft lead ball big enough. Could try to find a 45 cal muzzle loader round ball; they're usually $15/100 and are soft enough for slugging. I cast my own .435" round balls for kitten fart loads in 44 mag so I use those.
You'll need to expand your case necks to a thou or so under the actual bullet diameter. Lyman M dies are designed for this; they expand the entire neck. You'll also want to flare the case mouth. A big problem with 44-40 brass is it tends to be very thin compared to other cartridges so it'll crush easily if the bullets aren't seated correctly.
I wouldn't use hard cast bullets for a 44-40 since it's meant for black powder and soft lead anyway. A soft to medium alloy would probably be best with black or smokeless powder.
You might have a groove diameter of .426", .429", or .430", or .431", or something else entirely. With a nice thick copper jacket you can afford to assume and not run into too many issues. With lead bullets you can't assume without running into trouble (tumbling bullets, barrel leading, etc.). Remington makes jacketed bullets specifically for the 44-40 and they are .426". Wikipedia says 44-40 is .427" in a perfect world. If your dies sized the brass assuming a .427" bullet it is probably sizing it to .425" or so and combining that with the thin nature of 44-40 brass I'm not too surprised it buckled with a .430" hard cast bullet trying to get in.
The Lyman Cast Bullet Handbook has a lot of great info for reloading cast bullets. They've got quite a few peculiarities that aren't present when reloading copper jacketed bullets.
For the actual sizing step the standard tool is a lubrisizer which is designed to size, lube, and also crimp on gas checks. These can run $200-300 or more and you need sizing dies for each size; they are not adjustable. There is also the push-through sizing method that Lee makes specific dies for that work in a normal press and only cost ~$20 per size (again, not adjustable) but they only come in 1 or 2 sizes per calibre. They make a .427", .429", and .430" but if your bore is larger (often from wear in older rifles) then those are going to be too small. I had a machinist friend ream me out a custom made sizer to 11mm (.433") but it actually sizes to .4345" or so. I have a 44 mag with a groove diameter of .4315" so I need bigger bullets. The custom sizer sizes to .003" over groove diameter which is good.
There are different considerations for copper plated bullets and powder coated bullets as well.
