Kettlebell swings. Although it is a versatile tool and it's easy to put together some solid circuits. Single arm over head press, turkish getups, clean's, jerks, etc. That would be my vote. Probably start with a 35# bell.
Rational being: when tire flipping you should be firing your hips, keeping back straight and tight, and working grip. All three of those key attributes can be hit with a kettlebell. And it takes up virtually no space. Plant it in the corner when not in use.
just my 2 cents.