If your goal is to have Drobo host more than 2TB of storage, just add more drives. Drobo will add more virtual volumes (drive letters) as necessary so you can access all the protected storage.
Now if you want to have one big volume and you initially set up your Drobo for 2TB volumes and you have a regular Drobo (not Drobo PRO, PRO FS, or Elite), then you need to reset your Drobo to use larger sizes, and therefore should copy/move all your data off beforehand, as reset is a destructive process.
Unless you are running Dual-Disk Redundancy, losing a disk at this point will result in COMPLETE DATA LOSS.
Trying to continue the copy while the rebuild is running will only put maximum stress on Drobo, and if Docchris is right that the data pull overstressed things (perhaps the PSU is marginal, difficult to tell), then doing the data pull during rebuild is highly risky.
If you already have a backup of your data, then I would go the “just reset” route - you’ll have to copy the data back regardless, and moving it off it just a waste of time.
If you do not have a backup of your data, I would wait until the rebuild completes before continuing your data migration. Then seriously consider some kind of backup strategy. One copy of your data is NOT enough, no matter how safe/secure/whatever that copy is.
You say the remaining data can be done without… so I guess it’s really up to you.