Funnily enough I've been thinking about doing exactly this, i.e. using BH Auto Foam as the pre-wash snow foam (as per usual), but then experimenting with using BH Auto Wash (i.e, the regular shampoo that you'd mix up in yer wash bucket) diluted (of course) through the SF lance at a decent PIR, then leave it on the car and wash using the usual 2 bucket method. Need to muck about with dilution ratios, but good to hear I'm not crackers thinking about doing this, Lol.
Thought came about after using Auto Glym's Polar Wash (shampoo dispensed via SF lance - decent winter wash method when you can't be completely *****, but I'm now demoting this to a wheel wash method, as it's not as good a shampoo as BH AW).
@Simonwhite2000 + as many thumbs up as I have (that'll be 2 then) for BH Auto Wash as a shampoo for any coating you have on your car. Not the most 'suddsy', but it cleans VERY well, is PH neutral and does a good job of refreshing any coating you're using to protect your paint.