My recommenation is to rent a smoke machine from a theater or party supply house. The idea is that you need to make a smoke that is not irritating or caustic to actros or the audience. The smoke machine uses an oil that is nearly totally hypoallergenic, and has little if any odor.
For theater use, smoke is for visual effect, unless you are producing a religious scene, in which the smell of incense is part of the set.
If you let me know your location, I can help you to find someone who rents the smoke machine.
I understand your dilemma, however there are small water based smoke machines that will fit in the garbage can and use self contained water (a few cups).
There are also some oil based machines, that are also portable and will fit in the trash can.
I am sorry that you think there is another solution.
Any smoke has to be made with atomizers or heaters of some kind, UNLESS, you are willing to subject your actors and audience to allergies and respiratory problems.
Items that cause smoke that is least toxic are palms like those used at easter time in the Catholic Church (Good Friday) or incense that is used in Catholic Mass.
These have to be burned and create a fire hazard. IN addition, the incense creates an odor that may intefere with the production.
Of the non-machine solutions, I prefer the incense because it does not burn with a flame, and so it is safer. However, the odor may interfere with production.