What is the relationship between mountain breeze and air pressure?
There is nothing to reconcile: the higher pressure is where the cool air is. The weather report shows pressure fronts, and the low pressure ones are the ones heated up by the action of the sun.

If the mountainside warms up, the air in contact with it warms up. Warmer air has a lower density and it wants to float upwards, like a hot air balloon on a cold day. Meanwhile, surrounding air not heated up by the mountainside flows into where the heated air was, going from cool to hot. The reverse happens when the mountainside cools down: air gets denser and it wants to sink - when it hits the mountainside it just follows that down.

In a vessel with a constant volume, the higher pressure is at the hotter gas. In the unlimited volume of the atmosphere, the gas can expand as it pleases.