Just take hand luggage. We always travel hand luggage only and have done two month trips hand luggage only with two under two years also.
The only time we check in luggage is for skiing because of ski and board and ski boots size. But all clothing in handluggage still
10kg is quite a lot of luggage tbh
Tbh everyone always takes far too much travelling.
Just all wear your actual shoes, long trousers, and jumper/ jacket on the plane, and just pack flip flops and a few changes of clothing. Most holidays my 6 year old spends 95% of the time in just swim trunks or naked anyway so that saves luggage! The swim trunks just get rinsed and reworn next day.
A packing list for 10 days in sun for 6 year old here is something like:
Wear on plane: chinos, shoes, tshirt, casual blazer
Pack: 1 cardigan, 2 shorts, 2 swim trunks, 2 shirts, 5 tshirts. Underwear. 1 pjs. Crocs.
That covers day trips, days at pool, fancy restaurants in evenings etc. Most things can be reworn as only actual clothes on 6pm onwards. Otherwise rinse a tshirt or two in sink if needed.
I can fit my clothes plus x2 children's in one handluggage size suitcase for the week. Usually now they take own child's size rucksack with a jumper in, drink and some entertainment.
Buy suncream in boots after security if needed.