The first DLC is not scheduled to come out until June 2020, so you are not missing any additional content. And as far as I am aware, the customization options are not version exclusive.
To purchase different clothing, you need to visit apparel shops in the various towns in the Galar region. Each shop has a different selection of shirts, jackets, shoes, etc... so you will need to visit each shop to see what they have to offer. You can also check the stores inventory online on Serebii
In close proximity to the clothing stores are hair salons. Hair salons is where you can apply different makeup options, change your hair style and color, as well as changing the style of your eyes (as in adding stars to them).
There are 5 cities that have these available:
After you change your characters clothing and style as you see fit, you can customize your league card at any Pokecenter by interacting with the Rotom PC. When customizing your league card, you will be given various card backgrounds, effects, frames, expression, poses, and foils to apply. Each option has different requirements to unlock, all of these which can be found on Serebii as well