Will a 1000 watt inverter...

Inverter ratings can be very deceptive, and sometimes omissions and outright lies of various types. A reputable brand (someone mentioned one that worked for them?) is better. At least there is a big company to call and someone on the phone to ask questions of how big an inverter of their design is really needed. I hope.

Any of these comments could point to the trouble:
The cheaper-per-watt and more no-name the inverter, the more of these may apply.

Many cheapest inverters give only the peak power rating or maybe put the so-called continuous or RMS rating in fine print. The peak rating usually is twice the 'true' power rating. It's the rating that, if used for only a second or two, should not blow up the inverter or trip the breaker/protection circuit. Trying to use it constantly will cause quick overheating, perhaps destruction.

The unit may also have a 'duty cycle' which means that the unit is able to put out more power than is good for it, slowly and progressively overheating and cooking the board and devices time and time again, until it dies after a few weeks or months of what the user thinks is 'normal overnight use' according to its published specs. This is pure cheapness and often not mentioned, which is another kind of deception.

Additionally, the unit 's internal fan and heatsink combination may not be able to cool the guts above a certain power level, even though that level is below the 'true' published power rating, and this is not always mentioned, which is another kind of deception. It's similar to the duty cycle issue but worse because they cheaped out on the fan too.

Inverters are primarily rated for resistive loads. Anything with a simple transformer-rectifier power supply in it is a reactive load ("low power factor" load) and draws its current out-of-phase with its voltage while it is operating. Motors, as in fans and air conditioners and refrigerators are extremely reactive loads. This is not mentioned, but it is more a game of ignorance because inverters specs sometimes list the "power factor" range into which they will operate. Unfortunately most people don't understand its importance. Most equipment now states the power factor on the nameplate, like "PF 0.95" which is pretty good. 1.0 is a perfect resistive load. Get into the 0.8's, or worse like a sawzall or big drill where it varies all over, and it's harder and harder on an inverter.

Trying to buy appliances rated 'high power factor' is the best choice when using an inverter but they always cost more because they have more complex circuitry. Most "inverter" microwave ovens have good power factors. Always check the appliance data.

Some pics of my stuff;
decent power factor - voltage and current waveform timing almost the same:


Terrible power factor - the current and voltage waveform timings (peaks) do not match at all - high stress on inverters and some generators:


When I had the custom truck with an APU converted from a lighting trailer unit, I had voltage regulation issues with power factor and I added capacitors across the air conditioner motors to mostly correct this. This is well outside the scope of this comment.