Genes already cross between species. It is not as hard as many people think. On a biological timescale it is actually pretty fast.
Saying that something is done with GMO or Organic is not like writing pork on something. GMO is not something in a food and neither is Organic.
We do label food for contents.
Radiation and chemical mutagens will eventually produce a plant with a protein similar enough to the one on peanuts that causes allergic reactions. Labeling something as Organic does not change that in any way. It is a useless label and devoid of information.
All food should be tested. Our technology is advanced enough that all food should have DNA and protein assays stored on a government website. That way you could have a smart phone app and list what you don't want in your food (allergens, religious objections etc) and simple scan a QR code on the food and your phone could tell if the food is okay. Labeling GMO or Organic is never going to help you.
That you even call it a mutant organism indicates you really don't understand this technology and are acting out of fear. Sure we what do in the lab is not a conventional process but the methods we use in the lab are the SAME methods that nature fundamentally uses. CRISPR is taken from a natural source and all of this stuff we have learned from nature. Genetic engineering is darn safe and pretty easy to do now. The real danger is in radiation and chemical mutagens since they are more random.
If you create labels that you know are going to feed into panic and many of the supporters of labeling have a stated goal and making people react out of fear then you are not helping people. Labeling something as GMO is not going to help anyone and it is going to be used to create harm.
In the end I don't see the point in informing the public on an issue they are not qualified to make a decision on. If we want to have better food labeling we also need to have better education to go with that so people can make rational decisions.