Plumbing – What alternatives do I have to cap/plug the kitchen spray nozzle

Actually, it sounds more like a siphon or vacuum breaker working fine. It's not leaking with pressure on. When you turn off the sillcock and release pressure on the hose, it moves the water in the hose away from the sillcock, creating a vacuum at the faucet, which causes the siphon breaker valve to open. Then the water rebounds a bit and some comes out. They have only failed when they leak under pressure, or do not open at all. The brass cap thing on top of this sillcock is the built-in vacuum-breaker. The type described in Steven's answer seems more likely given where you say water is coming out. In either case, I don't see any sign of failure from your description of the behavior.

If the faucet set currently installed in the sink is sprayer compatible, it should have a third tube coming out of the faucet body/mixing valve. Also note, unless there is a sprayer shut off valve or a fitting that automatically stops any flow to the sprayer when the mating plug is not connected, it would shoot a hard stream of water out anytime the water was turned on. My guess is that there is no sprayer feed in the set you have installed now. If you put a pic of the bottom of the set from under the sink, we would know for sure.