Hi, Katherine.
to ATP.
We have a couple of wildflower meadows here that look good mixed with native grasses. Granted, our grass is different than yours; perhaps not as invasive in nature, but I think mowing actually helps grasses grow better in this situation. They regrow faster than most wildflowers and will get more light if other plants are cut away. Mowing the meadow once after plants go dormant might be beneficial, but hand-pulling unwanted plants would probably be best during the growing season. Toss in some seeds of plants you want in that spot after you do some pulling and those plants may fill the space, leaving less room for further intrusion.
Wildflower meadows are rarely grass-free however, so set a reasonable goal each season and in the meantime don't forget to set aside some time to just enjoy your blooms.