Fundamentally, yes, it is halal, but considerations must be made on exactly how the "loan" is contracted; attempting to run it like a modern bank where you're handling money that has been deposited (and can be withdrawn at will) would be questionable.
From reading through the proposal in OP's link, two viable scenarios come to mind, both of which are generally considered permissible:
- Mudarabah: This isn't really a loan, so much as investment in a business. One party (the rabb al-mal, in this case the "lender") provides the capital, while the second party (the mudarib, in this case the "bank") would actually run the business. The capital can be worked with in any manner stipulated by the business contract: As long as the contract permits it, then the mudarib may invest or lend out the capital on behalf of the rabb al-mal at her discretion.
- The rabb al-mal would be entitled to a share of any profits (if relevant) earned from the capital; exactly how the profits would be distributed would be included in the original contract.
- In the case of any loss, it would be deducted from the original capital. The rabb al-mal may end up receiving less than his full investment in return when the contract is dissolved.
- Bay al-dayn: This is, quite simply, a business loan. The lender provides a certain amount of capital, and the debtor (in this case the "bank") agrees to pay it back at a specified time (literally, this is selling an amount of money now for the same amount of money later). Until that time, the debtor can do with the capital as she sees fit — it's hers, after all; she technically bought it — including but not limited to investing it or lending it to a third party.
- The lender in this case is not entitled to any profits from the loaned capital.
- It becomes incumbent on the debtor to repay her debt in full when it becomes due; even if losses were incurred, the lender takes no share thereof.
As long as you're not attempting to work with more money than you actually have available (e.g. fractional reserve banking, which is common among modern banks), the issue of "selling things you don't own" doesn't even come up.