Put it simply, the transmission is the gear box, and the clutch is the part that engages/disengages the transmission to/from the engine. There’s also the shifting mechanisms that selects the gear in the transmission. The problem no longer sounds like the clutch or the oil. Sounds like the problem is in the shifting mechanism (spring, barrel, etc.) or the transmission.
Aren’t you in the SF Bay Area? For technical questions and suggestion on mechanics, BARF (bayarearidersforum.com) has many very experienced motorcyclists and home mechanics. For a bike mechanic in San Jose, I’ve used Sporttech Cycles. For even more specific technical question and service history of a specific bike, you should find the model specific user forum and ask there.
I woudln’t ride a bike that’s stuck between 2nd and 3rd gear other than to get it off the road. First off, starting on 2nd would be pretty hard on the clutch because you’d have to slip that clutch a lot at high revs to not stall the engine. Second, you may aggrivate the existing problem. Most importantly, it’s not safe to ride on public roads when the bike is not fully functional. You’ve got plenty of warning signs, so get that bike off the road and get it serviced. Don’t tempt fate and risk a major mechanical failure in the middle of a ride.