Double single-transistor logical inverter not working as expected
In effect, the base emitter region of Q2 acts like a forward biased diode: -





This means that you will only see around 1 volt maximum at Q1's collector.


  Is there any way to remedy this, and still get a 0/5V signal at both
  point A and point B?


Use a resistor in series with Q2's base - it has to be at least ten times greater than Q1's collector resistor. Or maybe use an N channel MOSFET instead of a BJT for Q2.