Yeah, Acronis True Image is the tool you need. You may be able to find a free Trial Version, don't know. There are a few others, though. In fact, LaCie's website may have some free utilities, so check there, too.
You say the La Cie has 2 x 1 TB drives in one box as a RAID1 array of 2 TB. Does not compute. 2 x 1TB drives in RAID0 will show up as 2 TB, but personally I think RAID0 is too risky to use because any failure in one drive means ALL your data can be lost. On the other hand, 2 x 1TB drives in a RAID1 array is a mirroring system that provides some security of backup, but it only appears to be a 1 TB drive. It's not as good as a proper backup system, but it is pretty good for day-to-day use. If that's what you really have in the external USB unit, I am sure your computer does not know anything about its RAID structure - it is just one 1TB external drive.
What you really want to do is create on the external drive a complete clone of your current C: boot drive that contains absolutely everything including all the hidden files. And eventually you will want to re-clone back to a new drive. There is a logic problem in all this we'll get to (next paragraph). Acronis True Image does this very well. Look through its menu system for the features and setting below.
The logical problem I see is: how do you boot without you current hard drive? Normally, Windows can handle an external USB drive because you have loaded USB drivers as part of Windows. But if you don't have Windows in a bootable device you can't reach the USB drive! Now, your computer's BIOS may be able to boot from a USB device. In that case you can ensure that, on the LaCie unit, the Primary Partition is a bootable Partition containing Windows, and you load and run from there. If you have that ability, your plan will work well. If not, you'll have to find another route.
There's another question i do not know. You will need to check the LaCie documentation on this. Plain windows does not know how to boot directly from a RAID array under its own control and needs particular drivers installed to do that. But in this case the RAID array is controlled within the LaCie unit. Does it really make itself look exactly like a regular IDE unit so that the computer has no idea there's a RAID anywhere? If so, things should work.
For now, let's assume you can set your BIOS to boot from a USB device and it does NOT appear to be a RAID array. So you install True Image on your current C: drive and run it from there. You use it to make a complete clone of your current C: drive as the SOURCE DRIVE onto the LaCie unit as the DESTINATION DRIVE. In doing so you must make the Destination clone a bootable device in the very first Primary Partition. You will have the option of adjusting the size of this Partition. It needs to be larger than the used space on the current C: drive - not the full 200 GB, but enough to allow making larger clone copies later (see below). If the software asks, you should choose to use the NTFS File System. You run it, then check whether the clone exists on the LaCie unit. This probably will have the result that the LaCie now appears to have ONLY that clone Partition on it. You then need to use True Image or Windows' own tools in Disk Manager to create from the Unallocated Space on the LaCie unit one or more additional Extrended Partitions that can be used for data.
For the acid test you shut down, disconnect your current internal drive, reboot and go into BIOS Setup to tell it to boot from that USB device. Save and reboot and see - does it really boot and run from the LaCie? If so, you are ready to recover from the failure of your old drive. So, shut down and re-connect your old drive, then re-adjust BIOS Setup to boot from that internal drive.
We're almost there. One thing to prod yourself: the clone you just made will not change (you don't want it to for most purposes!). So, from time to time as your internal drive does change with use, you need to re-make this whole clone. Here is why you need the first clone to be larger than your current drive used space. When you make a new clone, the process will be to delete that old clone, then make a new one in the recently-emptied space, and you want to be sure that space is big enough and all in one continuous block, not scattered over the disk.
The final step is recovering from the failure you have planned for. Now you have to remove your old failed drive, install a new one, then arrange to boot and run from the LaCie unit's clone. The first task will be to do the cloning operation again, only this time you use the LaCie's Partition as the Source, and your new internal as the Destination. Moreover, on that Destination that must be a bootable Primary Partition, you probably will want to set its size to the full capacity of the new drive, and not just the size of the existing (cloned) Partition on the LaCie unit. When done, you shut down and reboot into BIOS Setup and re-set the machine to boot from the new internal drive.