Don't worry too much about ketosis. It isn't important. Sticking to the plan is important and is the only guaranteed route to weight loss.
For the record, fat won't take you out of ketosis. So, for example, cheese won't take you out of ketosis. But cheese contains rather a lot of calories and salt, so eating it would interfere with weight loss while keeping your ketostix nicely in the pink. So even if you fried your omelette in butter it wouldn't take you out of ketosis. But there are more calories in butter than Frylight - and calories, on this plan at least, do matter. If you're happy to use Frylight, go ahead. It'll only make a couple of calories difference. Alternatively, invest in a good non stick pan - and then you won't need any fat at all.