Erik,
Bikes are not allowed in designated wilderness areas, of which the Sespe River Trail falls within (the Sespe Wilderness). Only foot/stock traffic allowed.
That said, there are some great bikepacking opportunities in the Los Padres, like Sierra Madre Rd to Painted Rock Campground out by Cuyama, Buckhorn Road which more or less trancects the Dick Smith and San Rafael Wilderness areas from the Santa Ynez River out to Cuyama, and some bikepacking opportunities around the upper Santa Ynez River area, all the way back to Ojai via Juncal Rd.