Would two rows of 0.1" single header fit? The spacing looks too far off to me. Your diagonal trick works well, but you have to do 9 pairs of 2 pins each on the diagonal, plus a spare.
You could test your square peg / round hole question by putting a dummy load on the supply. Maybe try four 100 ohm 1/4 watt resistors and put them all in parallel between +5 and GND. That would be 25 ohm, 200 mA, 1 watt total (the resistors will get hot). Then see if the 5V supply drops noticeably, as measured by your power meter. If not, then the square peg / round hole is probably fine.
You could test your square peg / round hole question by putting a dummy load on the supply. Maybe try four 100 ohm 1/4 watt resistors and put them all in parallel between +5 and GND. That would be 25 ohm, 200 mA, 1 watt total (the resistors will get hot). Then see if the 5V supply drops noticeably, as measured by your power meter. If not, then the square peg / round hole is probably fine.