You could get complicated with customer functions but if having an extra column (potentially hidden) doesn't matter then this would work:
In cell D2 enter:
=iferror(vlookup($A2,$E$2:$E$20,1,FALSE),"x")
Drag the formula down.
Under each month you can then put:
=sumifs($B$2:$B$16,$C$2:$C$16,B$22,$D$2:$D$16,"<>x")
Drag the formula across
This assumes you can change the month format in either column C or row 22 to be the same, ie. Jan, Feb, Mar or January, February, March etc.
Written on the fly and not tested so excuse any minor errors...