I can think of two simpler approaches off the top of my head, but I also have two significant questions:
The first way is to modify your stated approach and use Nested Project to insert your transitions-only project into your current project's timeline. That way, you just click down into the transitions project and click back up into the main project as needed.
The second way is related to my first question, which is how did you modify the default Blur transition?
PD will let you modify transitions downloaded from DZ and (as you mentioned) allow you to create an alpha transition. Either way, once you're in the Transition Designer you can save as many variations as you like. However, I don't see how you could have edited the default Blur transition without using a 3rd party app (which would have certainly let you save the new version).
So the other approach would be to find a similar transition on the DirectorZone website, then download, install, modify and save it so it's available for all future projects.
My other question/concern is that I don't see how you'd be able to copy/paste transitions at all. They're not standalone entities like clips or titles, and they have to be dragged from the library onto the desired clips on the timeline. Maybe I'm missing something, but it would help if you explained your process so we're both on the same page.
If you were instead just
hoping you could use copy/paste, then the only real solution would be my 2nd suggestion.
YouTube/optodata
DS365 | Win11 Pro | Ryzen 9 3950X | RTX 4070 Ti | 32GB RAM | 10TB SSDs | 5K+4K HDR monitors
Canon Vixia GX10 (4K 60p) | HF G30 (HD 60p) | Yi Action+ 4K | 360Fly 4K 360°