A lot of FFR guys carry space saver spares in their trunks. It depends on whether you’re using bolt on or true knock off wheels, but if it’s the latter then ERA’s FAQ page says they sell a narrow spare wheel:
Harbor Freight sells a very lightweight aluminum jack with, IIRC, 1.5 ton capacity. The handle is removable and breaks down into two pieces. They’re pretty low profile, too. I have one that I’ve used for autocross and track days for over ten years and it works great. When they’re on sale at HF (which seems to be every week, if you subscribe to their sale emails), they’re as low as $59. I would get one of those, find a carrying case to keep it from sliding around, and carry it in the trunk.