Since you have the diesel engine it will work fine if you don't use it too long. Making a quick repair - oK. Installing a cage from scratch trackside - not ok.
Modified sine wave is no problem for most things. The compressor certainly won't care. The welder shouldn't care either. I wouldn't use and inverter on our automated orbital welder at work but not because the inverter would hurt it.
You will be drawing about 230a on the DC side depending on the efficiency of the inverter. If you connect it to the truck batteries don't draw them down too much without letting them recharge. Starting batteries REALLY don't like deep discharges. If you run an arc for 3-5 min then let the batteries recover for about 10-15 min with the revs up on the engine to minimize battery wear. Not exactly a convenient duty cycle but one that the batteries will survive.
If you happen to have the diesel with the Ambulance prep package (dual alts) you are golden. Their combined alts will make over 300a Not likely though as they are very rare.
You don't have to use a brick on the accelerator to keep the revs up. Just look up "Aux idle controller". It plugs in under the dash and lets you set the revs to anything you like within reason. It is particularly nice in cold climates in the winter for a quicker warmup while you wait patiently inside.
If you don't mind spending a bit more buy a couple of the Honda EU2000i generators and wire them in parallel. They are small and light. They would be perfect for this and have the advantage of doing other jobs away from the track. Two of them wired together have more than enough power to do what you want. The power is super clean too. Not the cheapest option though. It might make you want to sell your big heavy generator.
Have fun. We will be out at the Lemons race at CMP in February if we can get the car back together in time.