Single phase fridges are inherently difficult to start for several reasons.
For cheapness they usually use a split phase starting circuit as it is cheaper than a capacitor start motor. There is also a problem that if the start is not virtually instantaneous the compressor builds up pressure before the motor pulls into the torque curve and it will remain stalled.
On a stiff mains supply there is no problem but with any source with significant impedance the highly inductive load drags the supply volts down and it fails to start.
Even small fridges will sometimes flatten a 3kVA alternator.
With alternators it is usually possible to just add a power factor correction capacitor across the line. In extreme cases you may have to add a permanent correction and also switch in a larger capacitor to correct the pf during start when it is much lower than the run.
The same trick usually works perfectly well with sine wave inverters. For MSW inverters you may run into serious trouble, the instantaneous voltage rise makes them very unhappy to accept any form of capacitive load and you may end up with a pile of smoke.
There is some possibility that adding a start capacitor in series with the start winding will make the thing approximate to a capacitor start motor (the winding is wrong for this to be totally successful but it may help considerably)
The inverter will not object to this capacitor as it will be in a series resonant circuit and look more or less resistive. What the harmonics will do is not possible to predict. ( if you hit a harmonic crawl it will never start)
If you are prepared to risk the inverter then you may be lucky. Few of these MSW inverters have totally foolproof protection so you risk taking out the mosfets.
You may be able to add a starting capacitor and a small correction capacitor across the line ( bigger MSW inverters may stand a few uF perhaps up to 8uF).
If you are nearly there now I would suggest that you try adding a starting capacitor in series with the start winding ( it will only be in during starting and will come out when the starting relay or switch operates).
I assume you have a 110v fridge so I can offer no suggestion of values ( our mains is 240V 50hz).
I think Nando is your best source of knowledge unless someone else has tried it on a 110v fridge.
If I had to make a guess, try 15 uf and work upwards. You need to aim for the highest voltage on the start winding ( resonance). If it doesn't start in half a second wait a couple of minutes for the pressure to drop before you try again
Flux