What's actually happening on that center pane
Your bay window's center pane is fogging in winter for one of two reasons, and they need completely different fixes.
Either warm, humid indoor air is condensing on the cold room-side glass (surface condensation), or the sealed double-pane unit has lost its seal and moisture is trapped between the panes.
The test takes five seconds. Touch the fog with your fingernail or a dry cloth. If it wipes away, it's surface condensation and a humidity issue. If the haze stays put, the seal is gone and the glass needs replacing.
Here's why the center pane specifically: it's the largest, it doesn't open, and it sits deepest in the bay recess where warm room air barely reaches. That makes it the coldest, stillest glass in your house.
Neither situation is an emergency. But standing water rots wood and grows mold, so you'll want to sort out which one you've got within a few weeks, not a few years.
Why bay windows fog before the rest
Bay windows are built to project out from the wall, which is exactly why they run cold.
The recess gets almost no heat from your floor or wall registers. Warm room air rises past the opening instead of flowing into the bay, so a pocket of cold, still air settles against the center pane.
Add curtains, a window seat, or a row of plants on the sill, and you've built a wall that blocks what little warm air might reach the glass. The pane gets colder still.
On the humidity side, indoor moisture has to go somewhere. Cooking, showers, houseplants, and even breathing add water to the air. In a shut-up winter house, that moisture finds the coldest surface and condenses. The center pane wins that race.
On the seal-failure side, the center unit is large and fixed. Large sealed units flex more with temperature swings, which stresses the perimeter seal over years. South-facing bays cook that seal in the sun, and it eventually gives out.
That's why, In typical service work, the center pane fogs or fails years before the flanking casements. It's the same age glass, but it lives in the worst spot.