My company is a foreign exchange trader with a number of clients. 
These clients lend their money to my company (this is not an
investment by any definition) and we use it for capital in our
exchange trading activities.  If, at any time, clients demand any
portion of their loans we accomodate within 24 hours.  The current
method for fund transfer between disparate domestic and international
bank accounts is via wire transfer.  Of course, wire transfers are
expensive and are charged at both ends of every transaction (sending
and receiving) - and it is very inconvenient since it requires
"in-person" attendance at a banking center and can't (at least so far
as we know) be done online.

We have looked into online transfer methods within banks, "outside of
bank" transfer options, and bill-pay services.  These all appear to
have limitations, mostly to do with either long delays (sending checks
through the mail) or severe transaction limits (under $5000).

We need to find a reasonable service for moving relatively large
amounts of money between accounts.  Amounts are generally under
$100,000 but our service is growing and we anticipate the amount caps
to grow above 100K very soon.

It sounds like a great way to launder money, hence the restrictions.

International mail need not be a long delay. Overnight courier and
second-day service is available from the large shippers.

Thanks for your honest observation.

Yes, we're very aware of the perception problem.  I can assure you
nothing can be further from the truth.  Our company is owned and
operated by principled Christian men, devoted to Jesus and committed
to uprightness (while still human and all too fallible).

Right now we use wire transfers, which are expensive and cumbersome,
but are not restricting in amount, frequency, or destination.  So if
we were laundering money, we would already be doing it successfully,
and using a common mechanism.   The question is not "are we money
launderers?" which we are not.  The question is, "is there a better
way to do what we already do, but for a much smaller cost?