Looking at how these work, I think these liquid smoke things need too much heat? I'm powering the thing from 4xAA batteries. I only need a bit of smoke...but this still might be too much current to draw.
Most likely 4 AA cells won't get you very far, unless they are LiPo or similar - something that can take a heavier current draw. Maybe NiMH would work...?
For the smoke - you might look into vape juice (ie - e-cigs - but not the commercial brands, you'll want parts meant for "box mods").
At its core, vape juice (and most fogger liquids) are combinations of poly-ethylene glycol and propylene glycol; for your small needs, just getting some standard vape-juice from an online vendor (or a shop) will work ok - just be sure to get the juice with zero nicotine in it. You can get it flavored as well (there seems to be infinite flavors) - which in many cases leave a pleasant smell, depending on the flavor of course.
In order to purchase the juice, though, you need to be of proper age to buy it (which will vary based on your local jurisdiction - here in the United States, there isn't a set legal age for vaping products, but most of the industry has settled on the same as for traditional cigarettes and tobacco products - 18 years of age).
Once you have the juice, you need a way of making the smoke. This is done with a simple heating element, called a "coil" in vaper parlance (there's a whole hobby of vapers of making custom e-cig equipment, winding coils, etc - its on par with the custom PC case modders) - which is made out of nichrome wire. This coil is either wound bare, or around something to soak up the juice - like a piece of cotton, sometime fiberglass, sometimes a small porous ceramic tube.
The way the e-cig or vape pen works is that this "wick" is soaked in the juice, and the wick carries it to the coil. When a button is depressed, current is run through the coil, which heats it up, causing the vaporization of the juice into a "fog" or "smoke", which is inhaled (thus imparting the nicotine hit, plus flavors if any - it is also possible to get a pure caffeine vape juice plus flavor, but there is debate on how well, if at all, such a juice works).
You would need to replicate this in some manner - you would want some kind of compressed air source to basically flow up and around the wick and coil, and a way to electronically control the flow of the compressed air (maybe the air comes from a pump, or a bottle and you use a solenoid valve or similar). Your code would first start the air flowing, then turn the current on to the coil - after a few seconds or so, turn the current off to the coil, then shut off the air flow. You need to keep the air flowing properly, and also regulate the current to the coil, so as not to burn the wick or the juice (which in an e-cig tastes horrible - and smells just as bad).
Note: If you are using LiPo cells - you will -definitely- want to carefully regulate the current; the nichrome coils present an almost dead-short across the cell or battery involved (the coils have fairly low resistance - sub 25 ohm is actually a high value), and you don't want to exceed the C rating of the cells if you can avoid it, because this can lead to damage of the cell, and ultimately to fire and worse (LiPo cells are not something you can be cavalier with). In fact, if you have heard anything about users of vaping or e-cig products having their e-cigs "blow up" in their faces (this is real) - it is almost always being caused by users who homebrew a custom box-mod system without any form of a regulator on the coil; this causes extreme stress to the LiPo cell(s) involved, which ultimately leads to the cells going into a "thermal runaway" mode, causing a fire, venting, exploding, you name it - in a device near the users face. It also doesn't help that the cells they use are sometimes cheap cells that are re-badged seconds from a reputable manufacturer, which are bought by the truckload, re-labeled (sometimes with higher C values than spec!) - and sold on to the hapless user (who may be box modding and not know anything about ohm's law - not that it would help the person, if the cells have been mis-labeled). This is also the same issue behind the exploding "hover boards" in the news as of late.
So - be very careful if you go the LiPo route - or avoid it, and go with NiMH cells.