Your drive levels for both 27MHz and control appear to be +/- 3V, which is not the way to drive a grounded NPN.
Both signals should be positive with respect to the emitter at all times.
You could either bias the transistor at -3v (rather than ground from the emitter resistor) or keep both signals between 0 and 6V (the base drive doesn't really need that much drive, though).
Note that a transistor inserted with collector and emitter reversed will still produce transistor action (but at a very low current gain).
The reason that the negative part of the output carrier is more positive than the original drive when the switch is ON (base is high, collector is approaching -3V) is because you have forward biased the collector base junction which will drop about 0.6V although there will not be much current in the collector (which is what the plot shows).
When the switch is off (base at -3V) the emitter will be at 0V under normal conditions, but the capacitance of the transistor is very probably transferring sufficient energy to the 2k emitter resistor to get you the output seen in the plots.
There will always be a bit of reverse conduction in this setup due to transistor output capacitance (a TO-92 package has quite a bit). This will give (as noted) a voltage divider between the effective reactance of the collector - emitter path and the emitter resistor.
I modelled it up in LTSpice and the reverse current path (from the 27MHz signal) is quite clear.
I ran this with a couple of different carriers (27MHz and 2.7MHz) to show the effects of junction capacitance.
Current in emitter resistor with carrier at 27MHz
Current through emitter resistor with carrier = 2.7MHz
To get close to theoretical response, apart from reducing the size of the emitter resistor (which you have done) look for RF transistors (therefore optimised for things such as junction capacitance). They are generally quite affordable.
Here is the result using a 470 ohm emitter resistor and a BFR183 RF transistor:
The BFR183 is 21p at qty 10 from Mouser (convert to your currency).
The HT12 encoder doesn't have a high current capability, so I would use a buffer such as a 74LVC06A between the encoder and the modulator.