Food for thought - when you have the spring bars engaged, the shank will be pushed UP in the receiver on the ball end, and DOWN on the shank on the truck end.
You will have practically zero up and down movement of the shank in the receiver when your spring bars are engaged. However, the amount of pivot involved due to the slop does indeed affect your initial spring bar adjustment. If you can't add any more washers to the head to get more tilt, then weld a bead on top of the ball end of the shank, and on the bottom of the truck end of the shank. This will give you more initial downward tilt on the shank (same net effect of adding washers to the hitch head).
I used some metal shims zip tied to my shank for a while, then finally had the weld beads added.
Shouldn't cost more than $10 to have a shop throw weld beads on there. Or less if you have a buddy with a good 220v stick welder.