On my third attempt, I decided to dust off the grey cells and use some brain power.
I decided to do the math. I measured the number of rows per inch, and then measured the front center edge. I multiplied the number of rows per inch times the total number of inches to see approximately how many rows I had. I decided to try picking up 3 stitches for every 4 rows again, 3/4 which is 75%, and I multiplied the total number of rows by this which gave me the number of stitches I should pick up. That made it a little easier because then I knew how many stitches I should have picked up halfway down the button band, checked my number of stitches, and then did the same thing on the last half of the button band. I was really pleased to see it came out well enough that I could be happy with it. Now to do it again on the other side. Now what were those numbers? Why do I always assume I can remember things??!