A tyre compressor with a football needle might just work indeed, especially for small passages and in small bursts...
What I always do is to soak everything in white spirit first, so that contaminations stay solved and floating instead of drying solid. Dirt that stays solid gets to meet one of my old toothbrushes, and only when all dirt looks solved or floating instead of solid and sticking, I blow the white spirit including dirt away with compressed air...
Rinse and repeat, just to make sure...
Also clean the inside of your airhose and airbox with a cloth and compressed air, since you've been driving without the clamp on the hose, I expect some sand and dust to stick around there, and that's probably one of the biggest enemies of a CV carb like ours...