The other problem with fog machines is they (most?) don't run continuously. Maybe go with a multi-head ultrasonic mister or a couple of those mosquito repellant coils that smoke.

 

There are "hazer" machines that are ment to run continuously. There generate thin haze and not smoke per se. The drawback is the price. They are used to generate haze for live performances, and thus are commercial quality, and have prices to match.

Another option is to run a timer on your smoke machine. This would allow it to operate in small bursts (3-4sec) at a time and every 10-15 Seconds. This should give you a light smoke effect.

 

You could use a few water mist foggers in a shallow tray underneath the grill.
I used three of these for a witches cauldron.

Amazon.com : AGPTEK Color Changing 12 LED Mist Maker Fogger Water Fountain Pond Fog Atomizer Air Humidifier with USB Adapter : Single Room Humidifiers : Home & Kitchen

Amazon.com : AGPTEK Color Changing 12 LED Mist Maker Fogger Water Fountain Pond Fog Atomizer Air Humidifier with USB Adapter : Single Room Humidifiers : Home & Kitchen

www.amazon.com

You can get them with or without lights should you want to add your own light effect.

 

For this application the water mist foggers would probably work the best. Since you don't need a huge amount of fog. Just make sure your prop can handle being totally soaking wet.

 

Just to give you an idea of how much mist you can get from three of those single water misters.

I used three in this application double side taped to the bottom of a tray that was deep enough to have 1-1.5 in of water covering them which is required for them to create fog.
The tray was on top of an overturned bucket inside the cauldron to bring it closer to the top and I had to tape around the outer gap between the tray and the cauldron so the fog would roll over the outer edge of the cauldron instead of down the gap.
Here's a better pic of the effect in daylight during a test run. They tend to use up the water over time, but thankfully the large diameter of the tray held enough water to keep the three foggers going all night. They also sell floats for these things so you can put them in a deeper bucket and not worry as much about running out of water. Though I haven't tried that method.

Overall I liked the effect. But I need to make a gasket out of a pool noodle or something to seal the outer gap. The black duct tape was an emergency fix.
Wind will also be a factor. On a very windy day no fog is going to stay put but you still will see wisps of it leaving your prop.
And since this fog is water based, as Cire stated, your prop will get wet as if it's been in a mist.
Hope this helps.

 

Helpful hints for this effect! Use a sharpie to draw grill marks on whatever body part is being grilled. Get a chefs hat for the skelly that's cooking and a string of orange lights in the grill helps light up the skelly at night like the glow of flames from an actual bbq.

 

I know that a fog machine would be too much... any ideas to have just a bit of smoke effect, or a fog/smoke machine that can be dialed way down?

There are small fog machines but they are quite expensive, there are DIY options for making your own using vaping pens, but it might be more hassle than it's worth... As said there are some timers out there for the generic lower-end fog machines that you might be able to dial into short-duration burst but it still might not be a steady smoke...

One option, since I presume it's in a real BBQ and likely won't be running 24/7 is to use long burn incense sticks, you could fabricate a simple heat shield/burn chamber out of two disposable aluminum turkey pans with holes punched in them, the heat from a few incense sticks will be minimal, especially if they are contained in a disposable aluminum burn chamber...

And last but not least, if you have the room to hide it and want to get creative, you might be able to do what I did last year for a DIY fog filled bubbles blowing over a fence with bubble cauldron sounds... I sourced a big cardboard box mine was about 3'x3'x3', I put a small fog machine on a milk crate outside the box and cut about a 4" hole in the side of the box for the fog machine to blow into... The fog machine was on an adjustable timer... I then cut another 4" round hole in the top of the box, and stuffed in some 4" flexible dryer hose, the other end of the hose I taped to the back of the bubble machine where the fan draws in the air to make the bubbles... In short, the fog machine fills the box with fog and keeps it filled with fog, the small fan on the bubble machine pulls the foggy air out of the box and produces a pretty consistent stream of non-stop fog... For your purpose, you could use an adjustable small computer fan to dial in how much fog to pull off and pump into the grill, and you can also scent the fog Froggy's sells a 'Charred Corpse' fog scent that might work 🤣

 

And a followup that I forgot...

Again, I'm assuming this is a real grill, so that said, here is a novel idea that will get not only some smoke but also some sound... That being hit up a resale shop like Goodwill and get a cheap VINTAGE iron (clothes iron you need an old school vintage one as new ones shut off if not being used) then Google up "drip watering system DIY" and look at the tutorials on how to make a drip waterer... Note you might even find using a real IV bag to be a perk as it can be part of the display, especially if you add some food coloring to the fog juice inside, note a little food coloring in this instance will be fine as it's not a fog machine that will clog up... Anyway, now that you have the iron (that won't shut off) and the drip system you will need to tinker a bit, you are going to want to mount the iron upside down so the hot end is up, now you are going to mount the drip hose above the iron... You will need to tinker with the iron temp and drip rate, don't crank the iron to high there is the small possibility that the iron will get too hot and cause the fog juice to turn into what could be some toxicity (IMO it's a small concern but still a concern) anyway starting with a lower heat and upping it as necessary while also increasing the drip rate of fog juice onto the iron... When you get it right you should get a sizzling sound with fog coming off it as it hit the iron... You might find that you get better results if you set a small aluminum foil 'cup' on the iron to help contain the drips vs them running off the flat iron surface...

Again you will need an old school iron that doesn't auto shut off from inactivity to do this, or hack a new one to stay on...