First of all realize that those cheapy inverters usually don't put out sine wave power. It may be a stair-stepped waveform, or just square waves in some really cheap units. Depending on the design of your strobe power supply, a stair-step or squarewave could cause problems, and not work properly.
Square waves may make shaver motors and heaters run okay, but can cause problems with transformer powered equipment.
Secondly, power is power. If you want to use high power (say 600 ws) much, be prepared to carry some big HEAVY batteries.
When a stobe goes off, it dumps the electric charge stored on a big capacitor through the strobe tube. When the strobe starts to recycle, that capacitor looks like a near short circuit to the power source for a few milliseconds and causes a big current surge. That surge can actually stall some cheapy inverters with little overcurrent protection. Not a good thing...
As far as power ratings, don't play too close to the edge. If you plan on using the full 600 ws, figure on a 1000 watt inverter to be on the safe side. Yes, it will be mostly wasted, but you will need the surge capability.