I had pals that worked in hospitals years back, and got some syringes and other items that I used for bike maintenance. I found using my automotive grease gun with a needle too cumbersome.
The syringes worked better, but as mentioned, packing them is a PITA and for me it ended up not worth the work.

My favorite tool is a plastic 'depressor' with one end with a 'french curve' shape. It's an OB-GYN disposable item. You can use your imagination. The shape is also ideal for bearing cups!
One end scoops, the other spreads a nice bead of grease.

These days, it's a tub o' grease, gloved hands and my 'depressor'.