Firstly, RAID shouldn't include your backups. Something catastrophic hitting one, will hit the other. Eg. fire, burglary, etc.
Ideally, you want to keep one set off site. Using the 2 disks you can keep one at a friends, and sync the up periodically. There are tools that allow you to do this. Alternatively, something like BackBlaze, Amazon Glacier, will give you huge or unlimited storage that is remote and protected.
Secondly, when setting up RAID you want to think about your priorities. Are you optimizing for speed, redundancy, etc.
I you want to have one unit in house, and one remote and periodically updated, you're probably better off with RAID5, which is a better mixture of redundancy and speed.
Finally, RAID won't allow you to upgrade disks singly.
Alternatively, using a something like Zevo's ZFS you can configure the Big5 as a JBOD and feed the disks into a ZFS RAIDZ. This will allow you to mix and match disk sizes, and upgrade over time. It will also keep better control of your disks, checksum all the data, and a host of other features.
Personally, I'm a big fan of the ZFS approach, as I think the performance and security are better than a plain RAID.
RAIDing using OSX software, on top of a RAID in the LaCie box is asking for some complex failure.
You're thinking of RAID-1 in the wrong way. RAID-1 is not an architecture intended to happily use a single surviving drive within a raid-1 group, and work out mirroring content later. Ultimately it will do that, but that is considered a recovery operation from a failed state, and it will destructively overwrite the mirror drive after replacement. In RAID-1, the two drives are expected to be both online and available all the time, as transactions will be committed to both. A loss of one drive will not stop the system from operating, but that is considered a failure mode, and as you've seen it will operate in a degraded state.
The expected user operation at that point is to replace the failed drive, at which point the RAID software will make a block-for-block copy from the surviving drive onto the new drive, a process sometimes referred to as resilvering. It doesn't look at files within the filesystem. This can be a resource intensive process, and as you note, can take hours. This is expected behavior.
If you want something that will enforce eventual consistency in drive contents, use something like Rsync. There's an app to help set it up. http://arrsync.sourceforge.net
Best Answer
I was unable to find how to rebuild the RAID from disk utility after a couple of hours of research. However, I managed to do it from the command line.
It will then kick you back to disk utility and start rebuilding the raid :)
Also, if the failed disk remains on the list, you can remove it: