Sand blasting and bead blasting are the absolute most unforgiving surface preps (that and a 32 machined finish that has been degreased) when it comes to sparks of any kind sticking to it. My theory is that the micro roughness of the surface creates an incredible amount of surface area for all those BB's and sparks to adhere (weld themselves) to. Ever seen how well grinder sparks stick to a windshield??? You can't get them off for nothing. (If any one out there has found a way, I'd sure like to know. I've tried acids, caustic, even buffing with rouge and soft cotton pads. Sux when you gotta buy someone else a windshield). Spark adhesion to blasted surfaces is just as bad. I'm betting that if you yourself were to grab the wand, there will be no better results. Try it if you think the washer person is failing in their duties. (As an inspector/supervisor, I've had to demonstrate for painters, grinders, hydro-blasters, welders etc that their poor technique was messing things up, and on some occaisions, have eaten a double slice of humble/crow pie. BTW it tastes like F%&#*ing - SH#T!!!!! especially when you screw it up worse than they did. Sometimes turns into a case of, yeah you're doing it wrong, but I can do it even WRONGER!!!!!!)
Can't remember the name of the product (I want to say "Spatter Free"? This was way back in the early 70's) but it was a gray powder that came in pry off the lid paint type cans, you mixed with water and it made a gray mud silty paste we painted on, and you best not have any in the weld zone, it would deflect the arc. It was the best antispatter next to acetylene soot and not near as messy.
On the other hand, if the paint is not sticking because of the anti-spatter compound residue, try pressure washing with "Simple Green" injected into the stream. Good stuff, we've used it as a pre-wash to get "black light" clean for O2 lines.
Might have to save the bead blasting for post weld/ pre-paint prep....or get some creative prep helpers with scrapers and small wire wheels.
Good luck

Joining in on this - you say that it is not the spatter that is the problem, instead it is the junk sticking to the anti-spatter that is difficult to remove?

Generally anything is removable once you find the right solvent.  Sometimes detergent and water works best and sometimes solvents like acetone work best.  One thing you might try is "Grafitti Remover".  I have no idea as to who makes it, or what's in it, but it has removed some stubborn stuff for us.  Spray it on, let it sit, wipe it off.  It might work if you put it on, then power wash or steam clean after.