Felt is convenient and cheap material. I like to make felt goods.
It may be difficult to cut a small piece.
Felt does not fray, but the edge is fragile.
I don't think there is a shortcut.
This is one of methods I do. It's rather good, but not very easy.
Draw or print all pieces of pattern on thin paper.
Cut them with margin. Make 1 or a few holes in each piece.
Stick it to felt with scotch tape.
Cut paper, tape and felt together. Remove tape and paper.