How would I make an adapter for connecting Atari Jaguar controller to SNES conso

The controller data is at NTT Data Keypad Super Famicom controller (NDK10)

Atari Jaguar controller is a 6x4 keypad matrix, reading it is easy. What I want is to take the keypad data and encode into 32-bit binary data (bit 12, 14, and 15 are always 1's, 13, 30, and 31 are always 0's), wait for the latch signal from SNES console, and send the binary data serially at each clock signal.

My issue is the timing. How would I make it work so it can read the pad continuously, and not miss the latch signal or mess the timing with the clock? The alternative is daisy-chaining 4 of 4021 but I would need 26 output from Arduino (on top of 9 to read Jaguar pad) and I would need to switch up to ATMega324PB just to have at least 33 GPIO pins. I would rather stick with ATMega328 as it's smaller if I can use 9 + 3 pins