You will find outdoor nomad catering offers: https://www.chateaudefontainebleau.fr/preparez-votre-visite/visites-chateau-fontainebleau/activites-dans-les-jardins/
Also take advantage of the many restaurants in the city centre: https://www.fontainebleau-tourisme.com/fr/preparer-son-sejour/manger/
It is possible to picnic in the English garden.