I don't have as much experience with hardware RAID controllers as I do software RAID, but it is certainly possible to create a RAID1 array in a degraded state under Linux's software RAID.
I would therefore assume that at least some hardware solutions would allow this. I vaguely recall reading about a particular controller being able to convert a single drive into a RAID1 array (without losing data along the way) as long as there was a small amount of space partitioned at the end of the source drive so the controller could store its array data without clobbering the end of a filesystem, but that was some years ago (Windows NT4/2000 era) and can't find the reference now (or remember which controller was being talked about, it may have been an IBM ServerRAID product as at the time I managed some machines that used these). This would give you your rebuild strategy (build the new system on a non-RAIDed drive, install the other drive as a degraded RAID1, copy the data, then wipe that drive and use the "single disk -> array" feature if present to rebuild by syncing onto that unit).
While you don't have spare storage of the sort of size needed for your array, do you have any smaller drives around that you could use to experiment on? Remove the current drives, build a simple system with the small drives and add some other data, then try your rebuild procedure and see if the controller behaves as you wish it to.
I would be inclined to take a full backup before starting any such procedure anyway, even if I'd tested the procedure on other drives first. If you already have a good backup arrangement then you should be able to just rebuild the drives without the extra messing around (and restore the data from backup afterwards). If not then this is an opportunity to take you first full backup, you will probably actually take less time this way around too.