First you need to determine which part of the inverter is causing the problem. There are two sections, the first is a SMPS to generate the high voltage DC rail (usually somewhere between 140-170V for a 120VAC inverter, double that for a 240VAC inverter). The second section of the inverter is an H-bridge to produce the "modified sine wave".
Check to see if the high voltage DC rail dips too much when you load it at 200W+ when the problem starts to occur. If it does, then the problem is in the SMPS section, if not, then the problem is in the H-bridge section.
Normally in this type of inverter the SMPS high voltage output is not tightly regulated, it goes up and down a bit. The output RMS voltage of the inverter is regulated by controlling the duty cycle: a longer duration of the positive and negative pulses results in higher RMS voltage.
A scope can easily tell what's happening all from the output of the inverter, as the peak voltage of the waveform will give you an idea of what the DC rail voltage is doing, while you can at the same time see what the duty cycle regulation is doing.
It's very unlikely to be failed FETs in the SMPS stage, there would definitely be some visual damage and/or blown fuses if that were the case. Those high current MOSFETs never go open-circuit without first releasing their magic smoke in a spectacular way! Bad capacitors or some type of fault in control/feedback circuitry is more likely. For example, I've repaired one where a faulty driver transistor caused one of the 4 H-bridge FETs not to switch on, resulting in low output voltage, reduced load capability, and a very unusual looking waveform at the output (more of a distorted square wave than a modified sine).