As you might know, the Munger Trail is mostly out in the open, by which I mean there is no shade. This is especially the case at the southern end in the area where you will start - White Bear Lake, Hugo, etc. and continues on into Moose Lake and beyond.
Depending on other weather conditions that non-stop heat from the Sun can take a toll. I don't ride the Munger partly for that reason. When I go north from Twin Cities I go through Princeton, Little Falls, etc. and on up to Brainerd and then take the Paul Bunyan Trail to Bemidji. I prefer that route but I end up in International Falls, not Duluth.
As far as your other question, it depends on the 7 year old. Those of us who do not know your son can't give you an accurate answer as to how he will do on such a ride, but bradtx had good advice in my opinion.
You don't say how far he has ridden but the picture I get from your "never ridden without training wheels" description makes me say, with no additional information - Yes, you are crazy to even consider it. I would want to see some expressed desire from him to ride any time he gets the chance before I would put him in such a situation of your creation.
I like your idea of a short camping trip as a trial. Bunker Hills has decent accommodations and is close enough and, to my surprise, is often not crowded. The admission fee is waived if you arrive by bike. When I last checked a campsite was $14 a night.
I would also schedule a test ride to see over what distance he can maintain focus. Maybe down West River Road from the 169 bridge at least to the dam and then back. I consider that road pretty safe because of the good shoulder but you might want to ride on a trail. Good luck.