178 (through the Kern River Canyon) *looks* like a shorter route, but it's very slow and curvy and taking it will add a lot of time to your trip. Also, there are stretches of 178 eastbound with rock "overhangs" very close to the road; one has to be watchful when driving a high-profile vehicle.
As commented, 120 will likely still be closed in April, and 50 is very far north; I suggest taking 58 out of Bakersfield to 14 (you'd have to check conditions over Tehachapi; it's only 4000', and should be fine in April, but who knows?), catch 395 and head up to Lone Pine. 395 is one of the best roads, in condition and scenery, in CA.
Any way you go, it's a long trip and there's really no easy or "fast" way to get there.