Hi
We have been cycle touring with our kids, similar age (ds age 10 and dd 8) had same dilemma last year and also did not want to really carry 2 tents yet, having said that ds carried tent and sleeping bag on his bike so getting a bit easier!
Last year we got a lightweight tent from decathlon:
link here I hope
I think we got the T3 ultralight as they didnt have the T4 ultralight in stock, originally they said it was only available in france but it looks like this year you can order it online. Looks reasonable size and only 3.5kg.
It was fine for 3 nights tho fairly tightly packed! Will do the same again this year and then from next year we will probably take this tent and our lightweight 2man tent for the dc to sleep in. The bigger tent means a bit more room to shelter from the rain.....
Where are you going cycling? Always on the lookout for good family routes?