How do I go about researching numbers stamped into Locomotive Bells to determine what railroad they may have been used on?

That's not an easy task. Most railroads stamped the locomotive number on the bell. You will need to review the rosters of the railroad(s) you think the bell was used on, and compare pictures to verify the bell style used on that type of locomotive.

In our museum we have a Santa Fe bell. I think it has 4 different numbers stamped on it.

... And some have hardly anything. I recently stamped ownership info into the bell on Soo Line 2442 in Marshfield, WI (now, there's a job that will really ring your bell) and the bell had exactly one four digit number stamped in it. Problem was, it didn't match the engine number, or the number of any engine on the roster. Turns out it was the Alco order number.

My son-in-law is the minister at a church near Louisville, Kentucky. On my first visit to the church, I quickly recognized their "church bell" as being a former steam locomotive bell. I had heard that some railroads stamped locomotive numbers on the inside of bells, but looking from below, I could not discern any number. I finally was able to make it to a spot where I could see the top of the bell. There I found the number 1875 stamped in two different locations. I believe now that the bell came from Louisville & Nashville class J4 2-8-2 #1875 (the L&N was the nearest railroad to where the church is located.) Hopefully some day I will be able to find out when the Mikado was retired to get a better idea as to when the bell was donated to the church.

Good luck in your search.

Les