This all really depends on where your data is stored.
If you put your home directory on the external SSD, will that solve the problem?
If so, here’s what I would recommend:
Setup your new Mac with your usual login account. Let it create your home directory “/Users/whatever”.
Create a second admin user on your Mac whose home directory will stay on the internal drive (you’ll want this as a safety net in case the external drive dies).
Log in to the 2nd account. Remove the “/Users/whatever” folder created in step 1 and replace it with a link to “/Volumes/ExternalDriveNameHere/Users/whatever”
Run migration assistant and hope that it respects the link you made in step 3.
If step #4 works, you still need to make sure that the external drive will be mounted before you log in. This command is supposed to accomplish that, but I have not tested it so I cannot guarantee it.
sudo defaults write /Library/Preferences/SystemConfiguration/autodiskmount \
AutomountDisksWithoutUserLogin -bool true
If that doesn’t work, then your best bet is probably to give up on the idea of using Migration Assistant and just copy your old home directory to your external drive.
Truthfully, I have to say that I give this fairly low odds of working, and if I was in this situation I would probably rather just spend the time doing a clean install from the get-go because I know it would work, rather than potentially waste a lot of time trying to avoid it. But it’s your time and your choice!
P.s. You can definitely install apps in ~/Applications/ although apps that get installed from the Mac App Store or from .pkg files will still go to /Applications/. Oh, and some apps will complain about not being in /Applications/ and ask if you want them to move themselves there, even if they are in ~/Applications/ (because they are trying to “help” and suspect that you might be running them from ~/Downloads/ rather than /Applications/).