You can book suites on points but for exorbitant prices.

For OP, the best way is as azepine00 suggests. Book your standard award room, then use $$ to upgrade to a suite. Often times the $$ cost is quite reasonable. Since you need a larger room, you will have to BOOK what you want, not hope for a kind free upgrade. That is the ONLY way to guarantee the room you want. Also make sure the suite will allow 4 ppl before you book it. Some hotels still will not allow 4 in a room.

Or you can book 2 connecting standard rooms. That may be better since some hotels will not allow 4 ppl in a room unless you have a suite (even then 4 ppl are not allowed in some hotels).

You didn't say where you were going in Europe. Rooms are A LOT smaller than the US. Often times, you can't put a rollaway in a King room (if you even get a real King bed which often not the case). In many countries, e.g., Italy, they are extremely strict on occupancy limits. Violating them can have you removed from the hotel, i.e., if you try to sneak one of your kids in.

For Europe, it's better to book 2 connecting standard award rooms.