Alright. Tried again now that 1.80x is out.
There is a tool named "gbata" that you can use to patch the ROM so it saves when playing on a GBA (and DS I suppose). The option is in the "SRAM Patcher" tab of the utility.
Now with that, use the Super Card's software to transfer the file, but make sure to uncheck "Saver Patch" in the ROM options there (just leave the "Enable Saver" checked). The standard save file provided by the software works OK.
Also, this is from my own experience with it, but leave "Enable Restart" checked as otherwise trying to save to the card will do nothing.
Right now the game is working for me, and I can save. The only problem is, I'm still getting the "Save file is corrupted. The previous save file will be loaded." message after the title screen. I can continue the game with no problem, but that message sort of bothers me...