#1, kudos for reading the labels!! You don't realize how rare that can be sometimes :)

Birdsfoot trefoil (Lotus coniculatus) is in the same family, Pea/Bean, as white clover, so you can use products labeled for control of white clover. Most lawn herbicides targeting broadleaf species that contain active ingredients such as 2,4-d, dicamba, mecoprop, etc. will have some activity on this species. Products with triclopyr are often recommended for harder to kill species, which can include white clover (e.g. WeedBGone Chickweed, Clover, and Oxalis Killer). The Turfgrass Weed Control for Professionals publication from Purdue University and others also recommends tricopyr for birdsfoot trefoil. 

Keep in mind that when the weather is very hot, dry, or otherwise the plants are under stress or flowering (as these here), herbicides do not always work with the greatest efficacy. The best time to treat perennials such as this is in the late summer to early fall when the plants are moving resources to the root system to prepare for winter. 

After you have achieved your desired level of control the work isn’t over. You need to do two important things. 1) Make sure you have a robust/vigorous stand of grass in your lawn to create a competitive environment for weeds. Practices could include re-seeding areas where the grass stand is thin (check herbicide label for appropriate timing if applicable), fertilizing the grass (early-spring and/or early-fall), watering when conditions are dry, and increasing your mowing height (~3.5" will help with broadleaves and stress management). 2) You need to be on the lookout for weeds emerging from seed. Some weeds can linger as seeds in the “seed bank” of the soil for decades! Controlling these weeds could include the use of a preemergence herbicide (e.g. products targeting crabgrass control) or manual removal of emerged plants. More information on lawn care from the MSU Turfgrass Team can be found on the following website: https://www.canr.msu.edu/home_gardening/lawns/