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.
Treatments: Komaza protein, Komaza Matani, coconut oil with sugar syrup. Hairveda Sitrinillah
Leave in: Fructis Sleek & Shine (old), Gliss ultimate volume
Styler: Umberto Giannini jelly, Boots curl creme, Rockin RInglets
Experimenting with: Mega-Tek, Curl Harmony reviver, Elvive Full Restore 5.