How to replicate MongoDB to a remote location?
Normally slow initial sync is because of slow network connection or VERY slow data disks. Have you measured what is transfer speed during initial sync.

Reason, why all data is copied with rsync, is that, that rsync thinks that too much has changed for delta copy.

And, yes option 3 is best solution, but only if you do that copy from snapshot or some other freezed data. Don't do it from live data files what are changing during copy process.