I replaced a grass yard with clover a few years ago and didn't have much success with any of the "cover it until it dies" methods.

Chopping the sod wasn't good for me either, so I gave in and used Agricel Grass Out a clethodim herbicide that only targets grasses.

According to what I could find, it isn't harmful to birds or bees once it dries. It dissipates in the soil in 3 days. It is slightly bad for fish but that's only a concern if you spray directly next to a stream/creek.

I looked at EPA reports as well, but the summary by the Minnesota Department of Natural Resources is easier to read https://files.dnr.state.mn.us/forestry/certification/pesticides/dnr-esra-clethodim.pdf

I know it isn't exactly what you where looking for, but if the other methods don't work out I recommend this.