I suggest one that's on a dimmer so you can adjust it at will.

I prefer a flush or semi flush light vs recessed. Stylistically this goes with pretty much anything and it won’t cast light shadows:

https://www.cb2.com/drum-flush-mount-light-19.75/s445942

Any suggestions on a warm overhead light?

I'd do recessed on a dimmer. What temperature (K) are the other lights? You say warm, but if your undercabinets, backlight, and overheads aren't all the same K it's not going to work well. I have 3000K in my entire house.

Do you actually get dressed and ready in the closet? That would change what works.

I find 3000K LEDs very yellow cast is that how you want to choose colors to work together ? Not me I like 4000K since it mimics bright daylight which is maybe the light you will be seen in .

Definitely use dimmers, but I'd do something a little fancier than recessed lights, especially if your closet is custom or semi-custom. Can you share renderings/drawings?

Agree on the dimmer and the bulbs (there are bulbs where you can change the color with an app and soft pink is often flattering) -

a fixture that has a milk glass or nice fabric shade could help too

Do you really need an over head light?

Select a fixture with and LED light source that adds no heat to the room.