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?
That sounds like an overly complicated setup with far more pumps than it should need. Or else you said pump when you meant tank in two places, actually, re-reading it.
In short, you can have as many pressure tanks as you like. Locating them closer to the point of use (the house) would be helpful, to a limited extent. That limited extent is the "effective volume" of the pressure tank at the house - while it's got pressure, you get better pressure at the house. As soon as it runs out of water and you become dependent on the pump, the pressure loss of the 300 feet of 1" PVC to the house comes into play, until you slow down your water use to the point where the pressure tank at the house fills up again.
If you do not change the setpoint pressure, the peak water pressure in the house will not be affected at all. Once the pressure tank at the house is depleted, the pressure will be the same as it is now. If you have a pressure gauge at the house now (or can add one) it might be useful to know the pressure at the house when "loads of interest" (showers, say) are drawing water.
Most pump controls can be altered to have a higher setpoint pressure, and most well pumps will provide that, within reason. General recommendations are to stay below 80 PSI - but if you are now running your pump on at 20 off at 40 (a typical default) you could turn it up to on at 30 off at 50, or on at 40 off at 60 - you will need to adjust your pressure tank (and any you may choose to add) for the higher pressure, particularly if they are bladder-type tanks - with the system drained, a bladder tank should typically be 2 PSI less than the low water pressure setpoint (ie, 18, 28, or 38 for the three ranges I've just given) and you may need additional pressure tank volume (because the effective volume of water a pressure tank can hold goes down as the system pressure goes up.) So, you can probably get more pressure without another pump, and possibly without another tank, but we'd need more details of what your system is doing now to know that for certain.
Unless your water use is extreme, 300 feet of 1" PVC pipe should not have a lot of effect on the pressure - at 5 gallons per minute, about 2.2 PSI - at 10GPM, 8.2PSI, 15 GPM, 17.4 psi
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