If he still wants to battle, you’ll need to work on his trust in you. Remind him that he doesn’t have to make any split-second decisions in a battle and that you and he are meant to work as a team in perfect sync. You may consider trying some team building exercises between yourself and your sylveon, perhaps by exercising with him or by performing two-person tasks with his help. The more you build up his trust in you, the less pressure he will (hopefully) feel on the battlefield.
If, however, he doesn’t want to battle at all, you may wish to consider having other pokémon fill the same role as he would and allowing him to operate within your team as a mentor to your battling pokémon, rather than a battle-ready pokémon himself. A number of pokémon can use fairy-type moves, for example, or moves that can cover the same types a fairy attack would cover. Granted, you would be hard-pressed to find a pokémon whose type combination functions the same way defensively as a fairy-type would, but on the other hand, part of being an excellent trainer is being creative about that.
Ultimately, it comes down to whether or not your sylveon wishes to continue battling. If yes, reassure him that he can rely on you. If no, then it’s best not to force him to battle, even if he would be a valuable member of your team.