Solution System:
Open the Equip menu, and these options pop up:
Equipment (Go to your actual equipment like normal.)
Automaton (Puppetmaster or sub PUP only, still functions normally.)
Vanity (Equip armor to change appearance, rather than stats.)

In the Vanity menu, you can equip an extra head, body, hands, legs, and feet, limited to equipment you can equip normally. This gear has no effect on your stats, but overwrites your currently equipped armor in appearance. Leaving the slot empty allows you to look like your regular armor allows.