I bought a cheapy hoping to convince Margaret to install one in the OKA so I could get rid of the last gas. It was labelled at 1800W At setting 9 (out of 10) it ran very well from the 1300W inverter, drawing 131A initially and then settled down to 118A (including a couple of amps of other stuff), even though it pulled the 9 1/2 year old 460Ah of AGMs down to about11.9V. They recovered within seconds of it being turned off. It boiled a litre of water MUCH faster than our Birko electric jug. BUT it lacked fine control at low settings meaning that maintaining a slow simmer was difficult. They pulse the power at low settings. Better ones may overcome that issue which was unacceptable to the cook.
Also have a cheapy and find it fine. For simmer, I have a very heavy based pan so the pulsing (which most of the portable units seem to do) has less effect.