Why doesn't white play Bd2 after this ...Qa5+? I was looking at an analysis which shows this position ( from Bacrot - McShane 2018). At this point Black just played ...Qa5+, and the only moves that are analyzed for white are Kf2, c3 and Qd2.

This is not the first time I see such a position, and I never understood why Bd2 is not even considered. As a black french player, I often discard moves like ...Qa5+ because it looks superficial to me after Bd2. But by looking at these games, it clearly looks like I'm missing something important here.

Any pointers ?

r1b1kb1r/3n1ppp/p1n1p3/qpppP3/3P1P2/4BNP1/PPP1N2P/R2QKB1R w KQkq - 1 10
The point here is that Bd2 blocks your queen's mobility and also leaves the defense of the d4 square. So after say Qb6 you pretty much have to go back with Be3, which will move closer to a draw by threefold repetition if blacks continues again with Qa5+.

Also if after Qb6 you play c3 instead of going back with the bishop, now your bishop is not well placed, and black's Queen on the other hand is very good on b6. So its safe to say that you kind of lost some piece coordination in the process.

One of ideas behind Kf2 is that you are pretty close to castling by hand with only two moves, such as Bg2 and Re1, and on the other hand black's Queen is misplaced on a5 and will most likely reroute later on to other squares.

All this being said, its still important to note that Bd2 is not a bad move. When i let Stockfish run for one or two minutes i see a diference of 0.1 for both moves. It gives 0.4 for Kf2 and 0.3 for Bd2, so its just an ok move.