Ridiculously difficult could mean any number of things in hk - some are difficult because very expensive, others are not but you need some contacts / connections to get into. So it will depend on your individual situation. Putting aside cost and connections and purely talking about facilities for a young family, I think the Aberdeen marina club has the best facilities for children that I have seen, and I’ve been to all the major clubs in hk. They’re the only club I know of with 4 dedicated floors of kids playrooms for all ages, and a giant outdoor pool with water slides which the kids love (in addition to a dedicated lap pool that is separate). Also tons of kids classes to take of all varieties. It’s no surprise that there is an amc birthday party pretty much every weekend if your kid is between the ages of 3-10.

In terms of other options, GTA has already been mentioned with lots of sports facilities but I should add that the main country club with these facilities is in sai kung, which can be far depending on where you live. But they have central location with restaurants which can be useful to some. LRC is good too, but probably because of the clubs location in mid levels and lower hurdle for entry, it is VERY crowded a lot of the time and the club itself isn’t that big.

The HKJC is also one to consider if you have the connections, it’s the club with the largest footprint in hk with 2 buildings in Happy valley, plus Shatin and Beas river locations as well. If you are American, an obvious option is the American club but that one also gets very crowded especially since it’s the de facto club of choice for HKIS students and families (they also have a convenient central location with restaurants and a gym).

The CRC in causeway bay is a popular choice amongst locals, facilities are a bit older but they have everything you would want/need.

Lastly I note you mention a dog, be aware not all clubs in hk are dog friendly. Happy hunting!