Hypothesis
Ideally, the ice and water should reach an equilibrium at zero celsius. But this equilibrium might take a long time to happen, based on the exact setup.
Looking at a typical setup of yours, the ice will float at the top of the container and there is water (but no ice) at the bottom. Water is densest at $\sim 4$ degrees celsius, and such water will sink to the bottom of the container. The water at the top is in contact with the ice and so, it must be near zero celsius, or getting cooler.
So I imagine a situation where we have ice and cold water on top, near the surface (which has to be in equilibrium with the atmosphere, but let us neglect that for now). There is also cold water at the bottom of the vessel (at 4 degrees celsius or cooler), with a temperature gradient leading to the top. The actual temperature profile will depend on all kinds of non-equilibrium physics having to do with the shape/size/conductivity of the container, the size of the ice cubes, etc.
My guess is that since you would have submerged the thermometer a significant depth into the container (maybe close to the bottom), you're measuring the temperature to be closer to 4 degrees celsius than to zero celsius.
Test
If my hypothesis is correct, then by stirring the contents of the container, you should be able to set up convection currents which will cool the contents more uniformly. You might want to be careful in not stirring the container too vigorously, for that will heat up the contents. I think gentle stirring should do the job without infusing much heat into the system, at least at the level of accuracy of your measurements.
This is a generic phase diagram :
![phase diagram](https://i.stack.imgur.com/DlOpK.gif)
I looked up the construction of a demonstration for classes cloud chamber using alcohol. For the air/alcohol gas line 3, constant pressure, change in temperature is where you want to work for your cloud chamber. Cooling it without condensation. The gradient of temperature in your chamber will define the thickness over the iced bottom where supersaturation can be maintained. So your choice of alcohol should depend on how slowly the phase diagram changes with temperature as to have a larger distance from the cold plate where the vapors will be supersaturated and tracks can form.
I do not think it is the gradient in the chamber that is decisive, but the one from the triple point to the vapor phase in line 3.The hot on top is to generate the vapour phase in the chamber. It will all depend on the phase diagram of your specific choice for vapor.
Best Answer
This link here describes how to build a cloud chamber without dry ice. Cooling is provided by an air duster ("The difluoroethane it releases is cold enough to supercool alcohol vapor, which is what you need for a cloud chamber.")