Yes, the same relay is used to drive the horn, or the pump. So if you want to use the pump you have to fit a siren, and get it T4
If you connect the headlamp pump to the wash pump it will just blow the fuse. The powerwash pump may not look much but the fuse for it is 30 A. You could use a relay to get around this though.
Also, as well as the number of times it operates, it's the length of time. The powerwash runs in a very short burst. You will empty your wash bottle very quickly.
I'll have a look around to see if there's such a thing as a a timer relay that might do the job.
Looking at the Haynes Wiring schematic, it appears that the Headlamp washers are linked to the Light Switch Module too, so the Washers only operate with the Lights on.
The wiring is also connected to the "K" Bus and Body Control Unit. I guess there isn't a simple way to wire this in, as from what I have seen, Cars with the Horn Alarm don't carry the loom for the Headlamp washers - So they need the Siren fitting and then wiring routing through for the Washer pump to all the necessary places. Unless I am overthinking this and the Horn alarm Plug and wiring is actually the same as the Washer pump wiring? The Plugs look similar from what I recall, but not 100%. If this is the case though, it would be a simple rerouting of that wiring along with a T4
session to reprogram the BCU
I am thinking too that the Volumetric alarm also means I will have issues wiring in my Dimming Mirror. But haven't looked into that yet as really don't fancy pulling off the Headlining right now :p