I'm unsure about how to convert the disk live, but I think rsync will be a better and safer way to copy your data over. It'll allow you to resume and keep the data intact in the event the transfer stops.
I did find a similar process completed by someone adding an external drive to their local system as an LVM. There's not a whole lot of information, but I think it will be enough to get your started:
"So today I discovered the awesome that is LVM. Installing Debian, I selected "LVM - Use entire disk". But the main drive was a slow and small 5200rpm laptop drive. Today I inserted my spare 1.5TB drive and booted up. Wanted the system on this bigger faster drive instead.
LVM approach: add /dev/sdc to the volume group, then run "pvmove /dev/sda". This moves all data from sda to other drives (only sdc available). No need to reboot, no need to unmount. While I'm writing this, the data is being moved.
Later, do "vgreduce megatron /dev/sda" to remove the slow drive from the volume group and voila. Data moved. (megatron is the name of the volume group and of my computer).
This might be old news to many but I just thought this was really cool :)"
Granted this was done locally, but I think with additional research, you maybe be able to accomlpish this.