[Exim] handling HUGE backlogs

We have a rather substantial backlog to a company who has been experiencing
severe mail problems as of late. By substantial, I mean, well, we have
*several hundred thousands* of messages queued up on a handful of machines.
The queues are massive.

The relevant behaviour of the remote site is such that their list of MXen
are often refusing connections, but every so often, one'll be able to
connect and send mail. Is there some way to have Exim stream as many
messages as possible down a single connection -- with the catch that the
messages have *not* previously been sorted using -qq, or that may have been
built up in the hints databases? I'd like to somehow configure a -qR@domain
run to use the same channel to deliver all messages, or at least batches
down the same connection. I suppose this may be similar to sendmail's
"using cached connection to host..."

Anxious to hear any suggestions.

Thanks!