"In an open environment, warm air has less density than cold air, therefore it has less molecules per cubic area, less molecules means less weight, less weight pressing against the moving surface of the barometric instrument indicates less pressure."

But pressure in the balloon has nothing to do with air in the balloon and everything to do with the weight of the atmosphere*.

Put a bubble of air at say 330' under water, the pressure of that air will be about 10 atm. It doesn't matter how hot you make the air, it will still be 10 atm. Though it will get bigger or smaller depending upon the air's temperature, therefore it's density will change with temperature.

Or put a barometric instrument in a balloon with air in it and put under the water, the air pressure will increase if balloon is put further under the water and decrease if it is let to rise in the water. The temperature of the air has nothing to do with the pressure; other than perhaps that the temperature may affect the barometric instrument- distorting it's ability to measure the pressure.

*If you heat the air, less air is needed in a given volume to give the same pressure as the outside pressure.
Or heat any thin walled container, cap the container, let it cool and watch the air pressure [14.7 psi] crush it.