Take your internal HDD and put it into the other MBP. Make sure it appears in the Finder. It may prompt you for the disk password or you may have to go into Disk Utility and unlock it the same way you did on your computer. If two "Macintosh HD" shows up - that is - your friend also has their disk named "Macintosh HD", then I would suggest renaming yours to "MacHD" for now to reduce confusion because it would be easy to confuse the two and then you might do something to your friend's SSD. In Disk Utility, select your "Macintosh HD", right-click (or Ctrl-click if you're using the trackpad) and select Rename.
Plug in the external HDD and see if it appears. You should do the rsync copy again but I need to look at the instructions because it will need to be modified to work on your friend's MBP. Another option would be if your friend has an app like CCC or Super Duper, you can use that. Or you can download CCC and install it if your friend is willing - there is a free trial period. I don't use CCC but it's highly recommended by people on macrumors.com and I've never heard of anybody having difficulty running the app.
https://bombich.com/
If you continue to want to use rsync - then do the following:
Make sure you know where the Terminal app is. It's in the Utilities sub-folder in the Applications folder - or you can use Spotlight to start the Terminal app.
You will need the flash drive to copy the ignore-files.txt file. You can see if can copy that to your external HDD or if not, copy it to your friend's home directory or Desktop - I would need to know where you copy it. Once you copy it, you won't need the flash drive.
In order to use the rsync command on this computer, because it's a working computer and not in recovery mode, I'm pretty sure you're going to need administrative privilege to do the rsync. So they would need to be there when the command is first entered.