Home built aircraft single pilot reported encroaching on SFO Class B airspace while maneuvering to avoid wake turbulence.

Narrative

I departed SQL for ZZZ via San Carlos's 'Oracle Departure;' and planned to transition the OAK Class C airspace to reach my cruise course/altitude. I had a passenger with me; another pilot I was bringing to pick up an aircraft from Maintenance at ZZZ. I had not flown with him before. I departed the SQL airspace and was in Class E airspace below the 1500-ft. SFO Class B shelf when I was handed off to OAK 'South Tower;' Runway 12-30. The OAK Controller instructed me to cross the 30 numbers at or above 1200 ft. I had to ask him to repeat this call because my passenger was talking and because this controller's transmission was muffled - loud; but unclear. All other controllers I worked with during the day were loud and clear. A short time later; I saw another aircraft passing directly in front of us; right to left; descending - airliner; possibly Boeing 787. As I was noting this; both my passenger and the OAK Controller called out as well. Concerned about wake turbulence; I turned slightly left and climbed; trying to stay below the 1500-ft. SFO Class B shelf while still passing over the track of the airliner. The OAK Controller called a second time before I was able to answer his first call. While climbing to avoid the conflicting traffic; I briefly exceeded 1500 ft. - track log shows 1510 ft. - and violated the SFO Class B. I still encountered some of the conflicting traffic's wake; but it was not severe. I was handed off the OAK North Tower; then to NorCal Approach. NorCal Approach notified me of a possible pilot deviation and provided me a phone number to call. My contributing failures [include]; failure to brief and maintain a sterile cockpit while transitioning complex airspace; [and] failure to assess all options to avoid wake turbulence - I could have made a right 360; but did not even consider it until after the event.Traffic arriving SFO at or below 1500 ft. leaves very few options for wake turbulence avoidance when attempting to transition the airspace east/west while remaining outside of Class B.Actions I will take to avoid recurrence [are to] brief and maintain a sterile cockpit while transitioning complex airspace; especially the east/west transition just southeast of SFO Class B via SQL and OAK or HWD airspace.Other actions which would improve safety [include]; publish an east/west VFR transition route or Special Flight Rules Area across the San Francisco Bay; which allows VFR traffic to self-separate from SFO and OAK arrivals; while avoiding the hazards of heavy wake turbulence and insufficient glide distance over water.