Yes that would be quicker. You also need a way to get the waste out of the cup.
You might be able to use an expanding collet Chuck or a pin Chuck.
Alternative 1
My wife used to turn a lot of tops for demos at fairs and such and give them to the kids that watched,
She used an expanding collet Chuck on a Morse taper. The blanks were prepped with a 1/4” deep Forster bit drilled hole. Collet fits in the hole twist the blank to lock it. Turn the top twist to take off the waste.
A quick search found this one. It is too large for you you can probably find one 3/4” size. The other issue may be that basswood is really soft and may not hold well on this Chuck. My wife used maple most of the time.
https://www.amazon.com/PSI-CXC-Expanding-Collet-Spindles/dp/B0006OC3EG
Alternative 2
a pin Chuck would work nicely probably need to be at least 3/4” long more waste.
The pin Chuck is round pice with a flat side and small steel steel rod the diameter of the flat cut away.
The rod is placed in the flat. The predrilled blank is slid over twisting the blank rolls the rod across the flat and locks the piece for turning. Twisting the other way rolls the rod the other way unlocks the so that you can remove the piece and catch to rod 9 out of 10 times the missed catch is searching through the shavings. I have one an inch in diameter similar to this one. This one does not use a pin. The end piece is cam mounted so it locks.
The 1/2” should do the trick.
http://www.peterchild.co.uk/chucks/pinchuck.htm
Option 3
A friend of mine used a wooden Chuck to hold squares for turning resets. You might consider making something similar.
The way it was made - three 1x3/4 pieces screwed to plywood disc on a faceplate. These made 3:sides of a square that fit rosette stock just a tiny tiny bit loose.
The fourth side was a similar piece ripped at a 15 degree angle with a 45:degree cut.
This made a sliding taper that will hold the rosette. The outside part is screwed in so that the whole piece will be too tight. The sliding piece was an inch or so longer than screwed on piece.
Blank put in the Chuck sliding piece got a light mallet whack.
Rosette turned
Sliding piece gets a mallet whack on the