Challenger 350 Captain reported encountering wake turbulence departing SAN that resulted in an inflight upset. Flight crew requested an offset to avoid the wake; but that request was denied.

Narrative

Enroute from SAN to ZZZ with AP (Autopilot) engaged. ATC cleared us to climb from FL320 to FL430. As we climbed from FL324 to FL340 the aircraft banked hard to the left between 45-60 degrees. Uncommanded. We both grabbed the yoke but the AP did not shut off and righted the aircraft without our intervention. We had hit the wake turbulence/vortices of the aircraft in front of us. We reported the event to ATC and requested an offset to get out of the wake. ATC denied our request stating the aircraft in front of us was 7 miles ahead of us so an offset was not necessary. Our TCAS showed the aircraft almost 5 miles in front of us. ATC then told us that aircraft would be leveling at FL340 and we would be climbing above his path. Once we climbed above FL340 we were above his wake turbulence and continued the flight without any other issues.