So I went ahead and did it today. It was pretty straightforward, and no more difficult than doing hub bearings or anything like that. I removed the thinner shim, which reduced play some but did not eliminate it. I'm going to ride it for a little bit to see if that's good enough. If not, I'll go back and play with the shims some more. I'd bet that just keeping the thin shim would work well. FWIW, the current small amount of play now resembles a similar 6 speed Tourney freewheel that I found in the parts bin that hasn't given me any troubles. Notes from the job for posterity's sake:

• Model: Shimano MF-TZ510-7, 7 speed 14-28
• Loosen the cone while the freewheel is on the hub. It is left-hand threaded.
• Outer (drive side) bearings: 30 x 1/8"
• Inner (non-drive side) bearings: 35 x 1/8"
• Shims: 1 x 0.014" (0.36mm), 1 x 0.005" (0.13mm). Thinner one was stacked on top (drive side).
• I used White Lightning Crystal grease for the bearings and paraffin-based sewing machine mineral oil for the pawls. I don't know much about lubricants, so don't take my word as gospel. Mixing grease and oil probably isn't a best practice.

7 speed, 14-28

left hand threaded cone

Watery inside, from just washing it

came with 2 shims

grease + oil