Not sure about any supermarkets near the lake or what you feel you need, but there may be some. However, the word 'supermarket' in Chinese often appears on shops that are too small to be called 'supermarket' in, say, the U.S.
Note that Dianchi, the large lake, runs for a distance of a few dozen kilometers south from Kunming, so wherever you are, you can't be close to every part of it at the same time.

Might bring own foods - I'm not sure what they might be in relation to allergy to nuts - but note that all markets here sell things that have no relation to nuts (e.g., meat, vegetables, fruit, fish, grains, noodles, etc.), and there are Italian nationals importing Italian olive oil and selling it through their own shops and restaurants (see listings), if you want to cook everything yourself.

Allergy. Bummer. Nebraska. 'Realities' of China. Poor food regulation. Caution. Wise. Bye.

Misleading sense of threat and mistrust perhaps also can be a problem, give rise to problems. Not unusual for this to happen when moving into an unfamiliar culture. Yeah, that's right, you can actually get things wrong - staying home can, one imagines, be a means to avoid danger.
My 'line' is that parents inform themselves and consider situation reasonably before decisions.
Geezer, you've made your contribution to this, why do you think mine is worthless?

Note: Thai rice and 'trust' - the thread is supposedly about food safety, not food preferences.
I wrote that fear is unpleasant. It is also usually disfunctional when in control, whether founded or unfounded; thus we attempt to calm down and think straight.

Best wishes, Annemompt - to put it simply, I'm trying to give you level-headed advice. You will note that people here, including foreigners, including westerners, do not all agree. Your decisions, obviously.

Priority of concerns.
Health of anybody.
Everybody has to bet, judge the odds for yourself.
Why trust Thai rice?
Should be greater attention to food regulation in China.
Use your head. Fear is unpleasant.
Couldn't find the word 'peanut' in the linked Wikipedia article. Yes of course it could happen.
Hope you get through the day ok.
My point has perhaps been made.