Lotta thoughts at play here.
Heating up the ring gear could certainly be done by hand but youd have to be pretty darn savvy at it to get it right without ruining the heat treat, thats assuming the parts are even heat treated. They may not be, depending on the alloys. The temperature to hit to influence dimensions can vary considerably depending on the alloys as well, same as the tempering temperature. I work with alloys all day long that temper at 800+ while others temper at 300. As long as you arent exceeding the tempering value by a considerable amount, the strength values of the metal shouldnt change by any drastic amount, the trick is knowing when youre there cuz you dont have the time to take a dimension before it cools back down.
We weld some heat sensitive metals that require being preheated to 700+ immediately before welding but are too thin to move from oven to bench and cant be placed in vermiculite to slow cooling due to purity levels so we weld them on a copper table heated from underneath to 750 degrees. Makes for a hot day of welding for sure.
I dont know that the ring gear is made of cast iron, kinda seems like it shouldnt be but its difficult to say since most any steel can be "cast" and not all castings are created equal, or even similar.
ER309, 309L, 309 ss is pretty similar to most other austinitic stainless welding rods. I dont actually think anything makes it unique for welding carbon to stainless, its just a run of the mill austinitic - 12-14% nickel, 22-25% chromium, some molybdenum, very low carbon content, decent levels of silicon and manganese certainly help but the nickel content is the usual suspect for bridging extremely dissimilar metals and 12-14% isnt that significant.
There are plenty of other welding rods that would perform just as well if not better for bridging carbon steel to iron. 312SS is one that I use ALOT for hard to weld crack sensitive alloys and dissimilar metals and should do a better job but its going to cost alot more, Inco 625 same situation, crazy high nickel and $$$. Actually, 410SS and even 4130 uncoated should also bridge carbon to iron fairly well but again those are going to cost more $$$. I just ordered a 10lbs bundle of 312SS from Lancaster and it comes in 275$/lbs.
I think I have a spare flex plate and ring gear, maybe i'll lop a piece off of it and run it through the XRF gun to see what its really made out of officially. Might give us more insight on what we are dealing with and how to go about fixing these.