Electrical – use a 1KW inverter along with a motor starter to power a water pump

I don't see induction motors as being particularly slow to start, so the answer is yes, it should be suitable.

There is no reason from a plumbing point of view why the booster pump has to be under the sink. It can be anywhere on the supply line to the kitchen sink. This may allow you to put the pump where it's noise is less of a problem. Oh, wait: If you are boosting the mixed water, it would need to be near. And if you aren't, you are going to pump water in circles through the mixing valve.

I dont see why an induction motor would be any quieter either. Noise may be due to resonance between the pump and the plumbing. This can be markedly reduced if you can use rubber hoses for the linkage instead of metal pipe. Mounting the motor on a resiliant pad can help too.

In general a more massive motor tends to be quieter, better bearings tend to be quieter. Induction motors may have better bearings.

Can the pressure in the whole house use a boost? Would a house sized booster pump and a pressure tank for the entire house be appropriate?

If you already own the generator I say go for it and try it out. If the generator is anywhere decent then it will regulate the voltage more than enough for a motor so you won't over power the motor with voltage. If the generator can't handle it you will not harm the motor unless you are able to stall it for an extended period of time; the generator shouldn't be harmed either it would be like trying to use a dead battery if it can't push out enough juice it just won't.

The reason I've notice that people suggest higher wattage ones is because they are typically built a little better and since they are capable of more wattage they respond to a peak significantly better, I'm going into peak discussion. Generators supply electricity on demand (even the power stations'), a household generator will produce approximately a few hundred watts at idle, as more items are plugged in/turned on demand rises and the generator senses it and runs harder. Since there isn't a preemptive notice about this, the generator will typically spike/peak higher than the demand and recovery back around what it needs to put out. A motor turning on with capacitors will typically be able to handle that lag, the capacitors will just charge up slower (almost not noticeable in most cases) and the capacitors work as rectifiers handling another other voltage changes during operation. Motors without capacitors also typically handle it but since they don't get a beginning zap from the capacitors you run a small chance of burning out your coils, imagine plugging a 220v motor under load to 110v source it will energize the coils but remain stalled.

If this is a more permanent installation (off grid scenario as opposed to hurricane prep, etc.) I recommend a separate motor starter kit - bad with naming here - before the motor or a heavy duty UPS/Battery system but for just when the power fails I think you'll be fine.