The only way to kill the dandelions and not most of the clover is to take a liquid broadleaf killer and paint each dandelion. There are various other ways to apply herbicide but most will kill surrounding plants. When your white clover gets going it should smother your dandelions down to just a few.

Any herbicide that will kill Dandelions will also kill the Dwarf White Clover. Dandelions are supposed to prefer soils that have a higher pH then grasses like, 7.5 or so. Some weeds can be controlled, fairly easily, by mowing high, but that, and your soils pH do not seem to affect them in the least and they do like the nice growing conditions we provide with that lawn.
If mechanical removal (digging them out) is not an option and the presence of them causes extreme stress then you may need an herbicde, but it would be better for your environment to not use one.

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.

Butyrac 200 commonly called 2-4-DB is the most commonly used herbicide for broadleaf control in clover plots. Pay attention to the DB in the name. This is NOT the same as the common 2-4-D herbicide, which will kill clover. Test it on a small test section first of course. Best of luck!