In a "permissioned ledger" (i.e., a distributed mirrored decentralized tamper-resistant database for a closed set of non-anonymous, legally bound entities), transaction processing would be done by the member entities, for whom the service would be compensation enough; and/or by external contractors, who would get paid in dollars through banks, the old-fashioned way.
They already have what you've described. It is called SWIFT.
Yes, I am sure that pretty good solutions that problem were known and used for many years before bitcoin. Those solutions may not be universally used for many reasons -- including inertia and risk avoidance.
Or safety. Delays of hours or days in interbank transfers are an important safety feature, and maybe exist for that reason alone. When instantaneous transfers are possible, bank hackers and money launderers often take advantage of them, by passing the stolen money through several banks in quick succession, to delay the investigators. Kidnapping and armed robberies also becomes easier and safer, since the ransom can be paid from the victim's bank account and cashed out before the police becomes aware of the crime.
As far as I'm concerned the delay isn't due to safety issues: since deposits aren't assets for banks but liabilities wire transfers transmit pieces of liabilities which the beneficiary bank has to accept, and to this regard banks have actual accounts open at other banks or, when this isn't the case, there is always a third party that does (which may be another financial institution or the central bank). This is the interbank market with fees (such as the federal funds rate, Euribor, Libor) pricing in the banks creditworthiness.
The transfer orders of the day from the bank A to B are packed into a sheet which is edited in the afternoon and sent to B which will consider it the next work day morning. In the meantime bank B redacts the sheet with the transfer orders from B to A. From time to time only the differences are settled between the accounts of the banks held at each other, or between A, the intermediary and B. If one party goes bankrupt it can't honour its account any more and this affects the solvency of the other party or parties.
There isn't anything even remotely similar to a distributed ledger in this process.