The very first thing you should do is Google around and see if someone else has tried to make any kind of save editor for this game before. Even if the editor can't change the name, if it can recalculate the checksum, then that would solve a big part of the problem.
The next thing I would suggest would be to dump the game along with your save data, load everything up in an emulator on a PC, create a savestate, and then try editing your name in the savestate. Unlike the save data, the savestate shouldn't have any weird checksum protection - meaning you should be able to re-load the edited savestate, save your game normally, and then write the save data back to your cart.
Otherwise, you're stuck looking for the checksum. You can easily find where the checksum is stored by comparing your save data before and after making one tiny, tiny change, like throwing away a single Pokemon or using a single item or something. There should be a sequence of bytes far away from everything else that will change.
But figuring out how the checksum is calculated is not easy. There are numerous different algorithms and it's not easy to tell exactly which data is checked. https://www.romhacking.net/documents/627/
provides an example for SNES Final Fantasy V. Some people would try disassembling the program to find the algorithm, but that's difficult. It is also possible to edit the program to disable the check entirely, but that's not easy either.