How do I get a lawnmower in and out of an old barn?
I would lean towards constructing an elevated platform inside the doorway. It would only need to be large enough to park and safely mount/dismount the mower. You could construct it like a backyard deck, perhaps adding steps down into the main floor area.

This would be relatively easy to construct and inexpensive, especially compared to anything messing with the existing foundation.

There are car garages in existence with wood floors and framing so I doubt the weight of a mower would be a concern.