I understand what you are trying to do. The link gives no technical information such as volts and watts. I will assume 220VAC and may be 15Amp for about 3300 watts. I expect a lot of problems with a relay caused by electrical noise when it opens and closes. I highly recommend using a solid state relay with zero cross. They work great with heating elements and the pump does not appear that big.
If you must use relays across the buttons you need to be sure it is low voltage as in a pool area open and unprotected mains voltage can be lethal. Unless you have experience with mains power and or somebody that does to help you I would NOT do this.
I can confirm that the control panel button feed 5v from each end pin but without bursting into the control panel I have no idea what the other pins in the ribbon do.
The 230v is under another panel.
Just a pity I can’t find wiring diagram as I think this would be possible
Try looking for a service manual and a replacement for the button module. The only thing I found was it had to be returned to the manufacturer or service station for repair including the cord. If you unplug it, pause for a few minutes and then plug it in does it remember its settings? You could cycle it with a remote relay if it does. I have seen many controls that are 5VDC or thereabouts but NOT isolated from the power line they did not use a transformer.