I'll put a disclaimer and say that I'm not 100% confident in this answer, but hopefully it helps clear things up. This type of question almost gets trickier the more you think about it because there are a lot of different things going on, and it's easy to get distracted by small things and miss the main points.
First of all, just because a force is applied doesn't mean kinetic energy is the main player. An object has kinetic energy because it's moving, not because it's applying a force. In this case, you're doing work on the bulb because you're applying a force and it moves some distance. Your hand applies a force to the bulb, and that comes from the chemical potential stored in your body. Of course the bulb and your hand do move, so they do have some kinetic energy, but that's not the main driver here, it's more of a necessary by-product of the fact that you're applying a force and doing work. That's maybe a subtle point, but I think it's an important one.
Now in the case of the cuff, I think elastic potential energy is the key player here. In a sense, all you're doing is compressing both the patient's arm and the gas inside the cuff. It takes work to compress them (work that came indirectly from your hand), and that work is stored as elastic potential energy, which appears as an internal force: the arm and the cuff "push back" on each other. Since neither is moving, there's no kinetic energy involved. Also, I'd like to point out that while elastic potential is present, it's NOT being "used up" to apply forces. That's something people tend to forget: force applied does not mean energy is being transferred (work). Something only does work if it applies a force and the thing it's applying force to actually moves. A box sitting on a table does no work even though it's applying a force to the table. Maybe you knew that already, but I thought I should point it out just in case.
Anyway, to answer your question simply: Chemical energy in your body --> Work done on the bulb --> Energy transferred to the cuff --> Work done to compress arm/cuff --> Elastic potential energy stored until you release it by allowing the arm/cuff to decompress.
Through the whole process, kinetic energy is present whenever things are moving, but it's never really the main driver.