Unusual voltage drop at BJT (2N2222)
BJTs don't work how you apparently think they work.  Remember that the B-E junction looks like a diode to the outside circuit.  You are seeing a 650 mV drop (5.00 V - 4.35 V) from B to E, which is squarely in the expected range.

You are using the transistor in emitter follower configuration.  You will therefore get significant current gain, but no voltage gain.  Actually the voltage gain will be a little less than 1.

The answer is to use a different topology.  A emitter follower simply isn't going to do what you want.