Mark, what strabes you are using? The analog strobe like the AB can tolerate voltage drop, and the Vagabond can handles 1200 ws of Bees connected to it; some digital controlled monolights don't work with inverter at all, some brand can not connect more than 1 strobe at a time.
I am only familiar with the 300W Samlex pure sine wave inverter. It can power my 2 Bees (about 500ws) with no problem. It has build in fuses for the AC and 12V sides, and a set of cable with clips that can be clamped on the battery posts.
I am with Skip with the safety aspects. Although the inverter current draw is lot less than the car's starter, modern car's electronics are very sensitive to voltage spikes, and often there are big fuse circuits that protect against shorts at the battery, blow one of those, then you are stranded. So I recommend to play safe and use a separate battery.