If I understand correctly, you have an antenna intended for a car body, but you'd like to use it loose in your house.
A solid ground plane will work, and it should be a quarter wave radius. For FM Radio, wavelength is about 3 m, so the ground plane should be 1.5 metres in diameter.
Instead of a solid ground plane, you could use four radial wires, 0.75 m long, extending at right angles to the whip antenna. They should all be connected to the ground bolt or metal base of the whip. You could use coat hangers or any other stiff wire - diameter and type is not important so select something that is mechanically strong enough. This will be a lot lighter and cheaper than the solid metal ground. It is what's done when mounting a whip on a fibreglass vehicle, or through a tile roof.