If they are 12V, then perhaps a VRLA battery on permanent float charge? You could use the smallest 12V VRLA and a tiny amount of float charge current, and it could easily provide the current required for the valves.
I have a garden watering timer that controls mains-pressure water and it seems to run for a year on a PP3 (but it only operates once a day and doesn't have to run a proximity sensor)

Think I've had something similar. Those types have a different operating valve. It uses a charge to open the valve. Once open the valve stays open until the close charge. Like a latching relay, only a water valve. That's why they last so long.

Had one that opened but then the battery was too weak to close it. Sprinklers ran all night until the morning when I found them still on. Had to shut the spigot off. When I got home from work I changed the batteries and turned the water back on. The sprinklers came on immediately. Had to do a test run to get the valves to shut off again. Then all was good for two more years.

Having sink spigot valves controlled by batteries is a good idea because you're working with potential voltages from mains that could accidentally find their way into the water stream and cause harm or injury. Or death. Nobody wants to be responsible for that. I DO like the idea of rechargeable batteries on a trickle charge. For as much as the valves are operated, a trickle charge could probably keep the power up as long as their is mains power. But perhaps the charger should also incorporate an isolation transformer just to keep mains potential away from the water stream.

 

Come to think about it - a 9V solenoid valve just might even work at 6 volts. But now you're closer to a low volt condition where the valves won't operate. Perhaps stick with a 12V battery and charger and have the charger on a timer so it only comes on during hours when it's very unlikely anyone would be using the sink.

 

It‘s no more dangerous than an instantaneous water heater, and they run off 230V, but plumbing has to be equipotentially bonded.

 

you need far more space than available on a functional workbench.

I use less than half a square-foot for my mouse movements, so doesn't take much space.
Your functional workbench must be pretty small if it doesn't have that much spare room.

But, to each his won.

 

It's quite large but there's no place to spare.

Get the missus to tidy it .

 

Get the missus to tidy it .

Then you'd never find anything.