Unfortunately you're stuck between a rock and a hard place unless your dermatologist can come up with another option. Salicylic acid is fat soluble so it's hugely drying, but the keratolytics that are water soluble will eat through hair as fast as they eat through skin, the chemical structure being similar.
Some people can moderate their psoriasis with diet modification and careful product selection, others cannot it's not an easy balance. It is an immune and inflammatory issue, it would be best to consider that in as many food and product choices you make as possible.
2a-2c, medium texture, porous/ colour treated. Three years CG, growing out mechanical and chemical damage = breakage and very high porosity. Past armpit length heading for waist.
CO-wash: Inecto coconut
Treatments: Komaza Matani, coconut oil, Hairveda Sitrinillah
Leave in: Fructis Sleek & Shine (old), Gliss ultimate volume, Inecto argan
Styler: Umberto Giannini jelly, Boots Essentials gel
Experimenting with: going back to basics