We want a play area for the children with a climbing tower and a trampoline, space for a rotary clothesline, possibly a small pool, a spot where a gazebo can be set up, a fenced area for our indoor-only cats, and a bit of lawn, so that everything is not so cramped. If the house now has about a footprint of 200 sqm (including garage).
then about 26.5 m long and wide
Yes, and then it depends on where the access is from. But that was already said?! From the south, one might want to increase the garden setback in front towards the street.
I would take graph paper and draw: building volume 10 x 12 (or larger as desired), your desired garages, the 3-meter distances... then separate areas. If it is financially feasible, around 800 to 900, but your wishes are also doable with 700. Drawing is fun - and that way you can already get familiar with the property.