Based on your own suggestion of an alternative to private property--that ownership be defined by usage--expropriation would be a form of direct action where you interact with available resources in such a way: If it is unused, you use it (expropriate it).
The thorny (and arguably moralistic or authoritarian) question of who are "the rich" wouldn't have to enter into the equation.
A key factor of expropriation--and a pretty key factor in direct action and anarchist activity generally--would have to be taking and not asking. If you run a project (well, business, really) that deals in relations of exchange with people as customers or tenants or recipients of loans, you are asking them for money. They can just go elsewhere for another price, so their unused resources won't be put to use unless they decide to (not unlike charity).
Also you are talking about a project that is an intermediary for redistribution: the resources are moved from point A (higher income people) to point B (those who don't have funds) through a mediating body (the bike shop). Expropriation would have to cut out the intermediary (e.g. people taking bikes for themselves or their friends/families/____).
So no. What you describe wouldn't be expropriation, it would be a kind of business negotiation.
For the record, a fair number of businesses and non-profits do run sliding-scale income-based services similar to what you are describing. I imagine they consider it a matter of fairness or redistribution or something similar. A logic that is distinct from expropriation.
On a different note, I don't really see how could someone loan money to, or rent to, someone who is much wealthier.