How to produce a simple beep/sound without using piezo-buzzer
At first, trying to use some side effect, as the sounding inductor core, you will end with very low efficiency.

RedGrittyBrick already pointed you to some existing buzzers as small as you need. Here is another proof link, that demonstrates that such buzzers exists.

If you really want to try something in DIY manner and is dexterous enough to work with 5x5mm electromechanical devices by hand, you can try to design some small electromagnetic speaker from a small solenoid and some thin iron foil as a membrane. Optional an anchor with a small hammer at the end that to knock on the internal side of the enclosure.

Another option is to open the smallest accessible piezo-buzzer and to try to cut only one part of it.

But all this is a very fine work and not everyone can do it. And is it worths?