We use these little buttons on everything! These Miniature Single Pole Single Throw switches have a good click to them and are breadboard friendly. Perfect as a tactile reset switch. These buttons are rated up to 50mA.
Unless you have a drawer full of batteries and are looking for a use for them surely this is doing things in reverse. I would think selecting a suitable switch first would be the logical step then finding the battery to suit.
I have seen a hell of a lot of push button switches of various flavours. Illuminated and not. LED or in previous times Lilliput lamps but I do not recall having seen one with a built in battery. Maybe they exist and I am just behind the times a bit but I can’t recall one. If such a switch does exist then I apologise and let us know. I am very willing to learn something every day.
Cheers Bob