I suppose the ultimate lightweight camping kit is a tarp strung between two trees and a bivvy bag. A bit spartan, though.
I would be less inclined to worry about weight, especially given the use of a trailer. If this is a two wheel trailer, e.g. Carry Freedom, Burley or similar, then my advice is let the trailer take the strain: you'll only feel 30% of the weight on the bike (ascending or descending hills excepted). When trailer camping, in my view one need not worry too much about shaving a kilo off the tent pack weight. I've successfully cycle camped with the wife and two young kids staying in inexpensive 5-man tents (or later a 5-man and 2-man combo) from the likes of Gelert, Coleman and EasyCamp. The sleeping bags were nothing to shout about either, but they did the job: you just wear some clothes in the bag if it gets cold one night. I should note I was towing all the camping gear in large Ortlieb bags strapped to the trailer.
I can hear the ultra lightweight camping enthusiasts recoiling in horror at this prospect. However I would say, if you have young children in tow, it's really not worth splashing out on an expensive tent, ultra compact sleeping bags, etc. It's amazing what kids can get up to in a tent: if it only cost £50 or £100 then there's no heartbreak if the tent gets unwittingly trashed by children in high spirits; nor is it a disaster when they traipse their muddy shoes all over the relatively inexpensive sleeping bags or sleeping mats, or spill their juice on them, or vomit over them, etc.
The problem of course (as noted above), is that 'family tent' to the industry means a bulky monster with three bedrooms resembling a bungalow, and weighing the same as a few bags of cement. There are though cheaper 'backpacking' tents out there which can serve as cycle camping family tents. Vango, Coleman, Gelert all do suitably sized tents. I'd be looking for at least a 4-man if it's two adults and a small child: inevitably 3-man means barely room for 2; 2-man just enough room for 1; and so on.