Water pump and inverter
The fridge motor is probably smaller than you think, the vacuum motor is a universal (AC/DC) motor so it inherently soft starts. The pump motor is likely a standard AC single phase motor and has too much inrush current for your inverter. When they say 1500W at start and 750W running, that's "marketing" speak because they don't want to scare people off. Starting an AC motor results in the inrush current being 500-600% of FLC for a second or two. But the power factor is very very low (i.e. .2 or less), so if you express it in "watts" the number looks better because W = A x V x PF. So 750W is about 1HP and at 115V single phase, that's about 16A FLC. That means at startup it can surge to 96A, but if the PF is 0.13, the W will be 1500 which LOOKS better than saying the starting current is 96A. For a utility connection it's irrelevant but for something powered by an inverter, it's too much and the inverter has a crowbar circuit that shuts it down to avoid damaging the transistors. Your best option (besides the generator one) quite honestly is going to be to get a different pump, either one powered by a DC motor that you run directly off of the battery (look for "solar pumps") or one that uses it's own inverter because that will come with soft starting. Or get a MUCH bigger inverter.