How to migrate from 13 inch MBP to 15 inch with 13 inch's SSD?
In the past, I have just swapped the drives in the two machines, and everything just worked fine.

Take the SSD from the 13" MBP and put it in the 15" MBP.

In the unlikely event that there are any problems, you can always swap the drives back.

EDIT: You might want to boot into recovery mode, and do a 'Verify Disk' in the Disk Utility, to make sure that everything is fine. As suggested in a comment above, consider a PRAM reset and an SMC reset, to make sure everything is on the same page.