Use LR to copy the picture files.
1. Connect and switch on the new disk.
2. Launch LR
3. In Library module, choose "Create new folder" and create a new folder on the new disk.
4. Drag-copy all your photo folders from the old disk to the new one, within LR.
Done. LR now knows where your files live on the new disk.
Time machine will realise that you have a new disk connected and will back it up automatically, starting at the next scheduled backup. If you don't delete the old photo files before then, you'll end up with two copies of each on your TM disk, which is probably not what you want. Expect the first backup to take quite a while! TM isn't clever enough to realise that the files on your new disk are the same as the ones on your old.
I did this when I bought a new hard disk. It worked like a charm, although my library was only about 15% the size of yours.
I'm not sure what you mean by "leave the processed file on LR". There is no processed file until you export from LR: there's only a source file (raw, or whatever other format) and a series of instructions on how to process it held within LR's database. TM will, of course, back up LR's database (although some have reported problems with such backup while LR is running).
Jeremy