Getting existing Time Machine onto a new, larger backup drive

The space on my Time Machine backup is running out, and I don't want to keep losing older backups. (Also, drives don't last forever, and I don't want to push my luck with an 7 year old drive).

I understand that duping programs are no longer able to clone Time Machine backups. So in a discussion with Apple today, the solution seemed to be to copy the oldest available backup to my current iMac, which can copy such files because it's an Apple device. That way there will be an "Old" image as the new Original Baseline, then everything that gets backed up after that will be the Comparison Snapshot, and we keep building from there. So:

1) Get the larger drive (new).

2) Move some of my data off the current iMac to make room.

3) Select the earliest available snapshot/image/backup from the Time

Machine

4) Copy that to my current iMac.

5) Hook up the new larger drive

6) Send that old backup to the new drive

7)Set new drive as the Time machine target

8) Run a backup to capture the current snapshot

The helpful support person said this would work, but I am not sure how to tell which of the files on my current Time Machine drive is the one I need to snag.

But before I start down this road, I wanted to get second opinions from the Community. Have I missed something I need to pay attention to/worry about? The overall goal is to have a new Time Machine drive ready before my new iMac arrives next week. (At which point the adventures with Migration Assistant begin)

Many thanks!