Yes, the tool in your photo is a type of budget lock opening tool, the key to the lock.
The Mark V used a C.2798 Key, Tee, for Budget Lock (Spare Parts Catalogue p.92).
However, the Mark V tool kit holds an “L”-shaped key rather than a “T” (or “Tee”) shape.
Keys of this type come in many sizes and tapers of the rectangular key ends. They come in “L” and “T” versions and some will work on the Mark V spare tire compartment and rear wheel covers and some will not. They were used in a lot of places other than Mark V cars. A short-handled “T” type will not work on the spare tire compartment, a “T” handle must clear the surrounding bodywork, while an “L” can do a ninety degree rotation without needing handle length for clearance.
The key you display in your picture may work fine. It is an “L” shape and is not one I have seen in Mark V tool trays before