Not sure if the reason you stated is the reason this fails.
you are right in the process, SnapMirror creates a snapshot and replicates that point in time copy of the volume, so the volume will grow as it keeps any changed blocks, however i would say that there would be a lot of change data (although i suppose it depends how long the initialisation is taking).
do you know that the volume fills and offlines breaking exchange.
what happens at the server level with this?
what you are trying to do, should work with very little problem, if the vol is filling, however just make the volume bigger.
but I would be tempted to look elsewhere, unless you know for definite that the volume offlines during this process.