Well, there's another possibility, involving bike racks on PAT buses. If the 78A isn't on a 1900-series bus, there's a fair chance it would have a bike rack. Mount bike on bus; go Downtown, and see if you can get an outbound 13A. I'd say get a 13K, but chances of that being on a rack-less 1900 are about 100% in the morning. The 13A gets you to the Showcase Cinema Park & Ride in McCandless, about 10 miles short of Westinghouse's new campus, but only 10 miles, about 1/4 of which is in North Park.
Briefly, you would bus to Showcase, bike to Ingomar Rd (same direction the bus was going), L on Ingomar, L at the next light and bear right onto the bike trail through the park, and L onto Pearce Mill Rd, which you will follow for several miles, until you go under the Turnpike.
Then, a choice: L onto Graham to R onto (really busy) Perry Hwy to R onto (really really busy) 228 to Westinghouse. Or stay on Pearce Mill to the end, L then R to past Treesdale to L onto 228 to L into Westinghouse. Or invent your own way to come up Franklin Road somehow. Translation: Nice ride until you get to hell (Cranberry).
There aren't that many outbound 13A trips, but if you can make the connections, it might not be an impossible trip.