I have a patch of white clover in my yard that I planted about 25 years ago. The soil is sandy. I don't remember the details of the planting, but I do remember that I broadcast the seeds by hand and lightly raked them in with a leaf rake. This clover is very well established and is the dominant plant in an area of about 300 SF.

Maifluer has the best solution given you don't want to hurt the clover. If you paint the centers of the dandelions with the herbicide, you should little or no damage to the clover, and if there is some, the clover should quickly spread to fill in any gaps. When painting it on, be careful not to let any drip from the brush onto the clover, and use a more concentrated solution than used in a hand sprayer. For example, if your directions say to mix 2 tablespoons of the weed killer per gallon of water, then mix it 50/50 or 40/60. When you put it on, you don't need to cover the entire plant, just dampen the center of the dandelion. Glyphosate will work well and leaves no soil residue that will hurt the clover. You can also use products containing 2,4-D or dicamba, although these can release vapors that may affect nearby clover (though it won't necessarily be killed). You may also need to make a couple repeat applications before completely killing the plants. Dandelions are particularly invasive where the lawn is thin, so be sure to correct the conditions causing thin areas to help prevent the dandelions from getting going in the first place.

"This is not a likely option because of the amount of area that my baby tears covers." It can feel daunting at the outset when the vista of work spread out before you seems like a bottomless pit, yes. But you simply start at one end and try to get into the zen of it and before you know it you have gotten a fair sized patch done. Repeat the next day and so on. Devote an hour a day and you'll be surprised what you can accomplish. Actually, weeding is kind of one of my most favorite things to do. It is of course tedious but the results are instant and very rewarding.

There are many species of clover, and they can prefer many soil types. There are two perennial types commonly grown. According to Wikipedia, Red clover (Trifolium pratense) grows from 8-31 inches tall, so it's not a lawn substitute. It has a deep tap root that makes it more tolerant of drought. It improves soil by fixing nitrogen and adding organic material as a green manure. White clover (T. repens) is low growing, forming mats as it roots along nodes of the stems. It tolerates mowing and prefers clay soil. There may be other better clover species and varieties that your local county extension agent can suggests. You should be able to find yours HERE.