I'm not clear on what's there. If it's a chunk of wood not attached to a tree and not buried in the ground, you can haul it off. I don't believe olive trees grow huge. If it's part of a root (sounds like it) then I would start with a pick axe and shovel and excavate around it. Two reaons: First to clear the dirt so you do less damage to whatever tool you use. More important: most root balls get small very quickly and you might find that by severing a few root branches, the problem is greatly reduced. You might find you could remove it with a bow saw.
As for an axe: a sharp axe in the hands of someone who knows how to use it is a marvelous tool but will still destroy the wood in a 90-degree cut. It will be damaged by the dirt and if I had to choose between digging dirt with a chainsaw or one of my carefully-sharpened axes, it would not be an easy choice.
Only if you tell the rental shop what you intend to do. Seriously - with a good dust mask and eye protection I would do it again. It left over 100 pounds of chips which I swept up and kept. I doubt that any of the dust was light enough to go anywhere This was in a boatyard where they applied bottom paint containing mercury and did all kinds of things which you probably cannot do today, but I don't think the lead made things any different.