Archery Individual - 2 players needed (each will play 1 match at the time)
Cycling Mountain Bike - 2 players needed (each will play 4 Assassin games at the time)
Equestrian Jumping - 4 players needed (each will play 8 Terminator game at the time)
Fencing Sabre - 1 player needed (will play 1 match at the time)
Gymnastic Artistic Vault -1 player needed (will play 1 match at the time)
Modern Pentathlon - 1 player needed (will play various game type but 1 or 2 at the time, this event will last longer to most of the other but shouldn't impact game load for the player can be engage into other sport)
Swimming Freestyle - 1 player needed (will play 1 match at the time)
Weightlifting Lightweight - 1 player needed (will play 1 match at the time)
If you want a different player for each of the sport during this 1st set you will then need 14 players, however I can see 1 player playing at least 2 sports since there is some that will play only 1 match at the time.
With that said you may use 7 players for this set, that will let you have minimum 3 players free for the next set, but that is if your team has 10 players only, in other hands if come up with 12 to 14 players you should have half of them playing every other set

.
The tournament may scary player(s) about game load due to the amount of sport but when you look closer and counting on the span of time of the event, then its not that heavy on game load.