The diverter in the pic is a Kingston Brass KBRP 112D and it is sold by
kingstonbrass.com
I got the center piston working better by cleaning it with isoproply alcohol, but,
I ordered a new part because I could see that some of the plastic is worn away on the bottom, and it might fail again. Total cost via FedEx is $25.
For an on-the-road fix for this problem, a practical solution would be to remove
the center cap on the faucet stem, and use a small flathead screwdriver or needle nose pliers to lift the diverter up and out of the faucet.
Of course, that is after turning the water pump off and pressure down. After putting the center cap back on, and the water pump back on, the hot and cold water will work ok.
The sprayer hose will be getting only partial water pressure, so mostly useless.
I don't know if this faucet was original or changed out.
And, if ever needed, a 1 gallon water jug can be trimmed to fit perfectly into
the bathroom sink for transfer of warm water next door to the kitchen sink.
A little something learned is better than nothing.