Maybe try an iron-chelate based weed killer? If you use it over the whole lawn, it helps to make the grass healthier and deeper green while nuking many weeds. I use it with 100% success on dandelions, but have no clue how Lily of the Valley reacts. Only issue is that if you use it in select spots, it leaves dark patches of grass.
I do a little bit of lawn restoration for work, and in many cases, when there is a bad weed/plant growing in the lawn, many home owners try to use band-aid fixes, wasting their time. Sometimes it's best to either rip everything out, or cover the lawn for a couple months during the middle of summer with black plastic after watering the soil super heavily. There isn't much of anything, even the seeds, that can live through that or go without sprouting and dying. Then pull the plastic off and do a deep de-thatching, then reseed. Sometimes you can get black plastic used from construction companies or private home owners who are looking to get rid of old stuff. If it blocks light and holds heat, it will nuke your lawn clean.
Thanks. Does something like that exist? I did throw some relatively high iron fertilizer this year. I'm pleased with how most of the lawn looks.