You have already identified the Super I/O chip, and found the datasheet.
The datasheet describes on p. 80 how to get into "extended function mode", which is needed to access the control register.
Bit 0 of register CR22 is called "FDCPDN", which translatse to floppy-disk-controller power-down, so I assume if you write a zero to that bit, then the floppy part will power down.
Other functionality is register CR30 ("activation control" for the logical device of the floppy, possible you have to disable this, too).
You can also use register CR60 and CR61 to set the base address of the floppy controller, so if this is exposed in your BIOS, you could also use this to move the floppy controller away from the port range occupied by the external controller.
If neither disabling the floppy nor setting the base address is not possible with your BIOS, then maybe this functionality is not included in your BIOS.
So finding a BIOS update which would include this functionality would be best (but that's unlikely).
You could also try to modify the BIOS yourself, which would be the next best solution, but that's a lot of work, as you first need to reverse-engineer the BIOS: Find the code that initializes the Super I/O chip, and modify the values written to disable the floppy.
In principle, you could also write a small assembler program that does it after you boot from the harddisk. However, that would mean that during power-up both floppies react to the floppy port addresses, which is not ideal, and may screw up the configuration process, so you also may have to properly initialize the floppy controller and corresponding BIOS RAM locations after your powered down the onboard controller.
All of these require that you are skilled in assembler programming.
I also had a look at the AHA-1522A manual, and it looks like you cannot adjust the base address of the floppy controller on that card (which would have been another way to resolve the port conflict).
In case you come across any port base settings: The first floppy controller is typically at
0x3f0, the second at