Here's what you may have to do if you can't get the iMac to boot and reduce it's data size to what the Mini will take.
1 - set up the new Mini with a user account that has the same full name and short name and password.
2 - get the new Mini up and running,
3 - see if you can connect the Time Machine drive to the Mini and access it. If you can proceed with Step 4.
4 - go to the Time Machine menu in the menu bar with the Option key held down and select "Browse Time Machine Backups". (If you can't browse the try holding down the Option key and selecting "Browse Other Backup Discs…"

5 - go into the latest backup, select the contents of the Pictures folder and restore (provided they will fit on the Mini).
6 - once the restoration is complete copy the Pictures folder to your external drive. Note: make sure your Photos library(s) work correctly on the EHD before deleting the contents of the Pictures folder on the Mini. Do not delete the Pictures folder itself.
7 - do the same for the Music folder.
8 - do the same for the Documents folder if you want to put it on the EHD to provide as much free space on your Mini's boot drive.
NOTE 1: it's wise to try to keep a minimum of 80-100 GB of free space on your boot drive to facilitate optimal system and application performance.
NOTE 2: I recommend an SSD external drive as it's is more reliable and faster than a HDD. I also recommend getting it (if you don't already have one) from OWC (MacSales.com). They are considered the premier hardware supplier for Macs.
If budget is a concern the the following setup might be worth considering:

Even though the above may not be as fast as an SSD in its own case it is quite fast: I can open a 56,00 image Photos library in 2 seconds on such a setup. I have several with my Mini.
I hope this helps.
