Beat whipping cream, crème fraîche, sugar, and zest in a bowl with a mixer until thick enough to hold a soft shape. Chill.
Cut cake into parallel slices 1 1/2 in. wide, then cut slices into 32 chunks, each 1 1/2 in. Save remaining cake for other uses.
Heat grill to medium-low (300° to 350°). Put strawberries in a large bowl. Microwave jam in a glass measuring cup until bubbling, about 30 seconds. Using a pastry brush, dab hulled insides of berries with jam, then brush remaining jam over berries and turn gently to coat all over.
Hold a pair of bamboo skewers so they're slightly separated and thread a chunk of cake onto them, then a berry crosswise, then another chunk of cake and another berry. Repeat to make 15 more kebabs.
Grill kebabs, covered, turning once with tongs, until grill marks appear, 3 to 4 minutes; cake should release from grate when it's toasted, but if not, nudge with tip of tongs. Serve with cream mixture.
Make ahead: Through step 4 up to 2 hours, covered and chilled.
Note: Nutritional analysis is per serving.