Raspberry Pi 4 with Mesa 7c81

The 7C81 only has the option of all pullups and all pulldowns
on its I/O ports.

Simplest solution would probably be to add a stiff pulldown (say 620 ohms)
on the relay I/O pin. There is a firmware option for fixed parallel port type I/O
but currently its global (not per port), This firmware makes the parallel port
output pins default to outputs instead of inputs at startup or watchdog bite
What firmware are you using?

so a stiff pulldown to fight the pullup of the mesa card for the inputs?

I am using 7c81_5abobx2d firmware with BOB boards on P1 and P2.

Edit:
I did a little bit of digging.
- The relay pin P17 is IO7 of the Mesa ports, which is on the edge of the pull resistor array. I could desolder the array and put it back on shifted by one to remove the resistor of the pin.
- On the BOB there is a 10k pullup array on the relay pin. but the signal comes through a via in the middle of the pullup resistor and the 74HC... I could cut the trace between the via and the resistor to remove the pull up on the BOB
- solder in a pulldown resistor on the BOB (if for whatever reason the flat cable between the Mesa and BOB gets bad).

Honestly, I would just add a  leaded 620 Ohm pulldown
resistor on the back of the breakout card and not make
any further modifications.