Dick, I am sorry to say that we are going to have to wrap this Thread up. It is not that I do not want to help, just that I cannot help any further.
Until or unless you can find a way to work between F2 or F12 and get a USB stick with Linux Mint on it (does not have to be 18.3 could be 19) to boot ahead of the hard drive, we are hamstrung.
For the moment, your backups by Timeshift appear to be going fine, and you will not run out of space with that partition allocated to it on the T5.
As long as your full install on the Toshiba continues to run OK, you are fine. If it goes pear-shaped, then you have problems.
You said in #6 on page 1
I don't recall the steps I used to setup the same system on my Toshiba notebook
... and so you must have been able to boot from a USB stick at that time, it's a pity the steps were not documented then (spilt milk), but it means the Toshiba is/was capable of it.
If you have that new SSD on order from the US, and you are going to be going to the Big Smoke to get that installed, then you might be able to get the technician to show you then how to alter the boot order, so take the LM USB stick with you.
If or once you are able to do so, then don't come back to this Thread (coming up on 11 pages, and too unwieldy), but rather start a new Thread in, say General Linux, on how to restore a Distro from Timeshift.
OK? Thanks Dick, and sorry I can't help further on this one.