I would look into a TSC (Thyristor-Switched-Capacitor). Plenty of articles on the internet discuss the basic topology.
Your problem is that the compressor motor is immediately switched on, which applies a highly reactive load to the generator. The only way to get reactive power out of the generator is by increasing excitation via the DC magnetic field, and this takes TIME. The overall time is approximated by a 2nd order transfer function consisting of the DC rotor field time constant (L/R) and the open circuit time constant of the armature (T'do). The load is applied immediately, yet the generator's armature output rises slowly, hence you end up with low voltage output from the generator for a good amount of cycles. Perhaps the generator catches up eventually, but not before your other connected loads have experienced the same voltage drop and are complaining or even dropped out.
Do not pursue synchronous motors or flywheels - you do NOT need extra watts (which a flywheel would provide) you need extra VARs. This is only possible with magnetic excitation and/or passive capacitance. The TSC is a passive solution, and allows cycle-by-cycle adjustment of the amount of capacitance. I suggest you might want 2 or 3 stages for optimal voltage regulation. One stage is pretty aggressive, and risks overexciting the generator when load drops. As well, a synchronous motor in theory is a proper solution for getting VARS, but it suffers from the same problem as the generator, which is the 2nd order lag transfer function. They are simply not fast enough to supply what is needed for an across-the-line motor start. Thyristors are there just as fast as you can detect the start and respond with an 'on' signal.
Last option would be a STATCOM, which is insane for a 2 kW system. Those are essentially PWM drives run off a DC bus which recreate the AC waveform. By adjusting phase angle, they are able to absorb or supply reactive power. Not impossible, but really quite the overkill. TSC is simpler and easy to implement with off-the-shelf products.