Can a Rabbi of a Shul claim back expenses for the outlay of food for mishloach Manos to give to his congregants?
I would strongly assume that the basic bare-minimum requirement -- two foods to one person -- is among the rabbi's personal responsibilities and therefore not charged to the community. It has to be given from funds that are yours to do with as you please. (Teenagers living at home, for example, should ask their parents "can I totally have this fruit basket? To eat it, burn it, do whatever I want with it?" And only after obtaining such permission is it theirs to now gift it.)
If the rabbi is expected to send lavish mishloach manos to a large contingent of the community -- and G-d forbid he forget any major donors! -- then I could certainly see that as something the community could be subsidizing -- seems like a reasonable discussion for the synagogue trustees. (And if necessary, I'm sure the Beth Din of America would gladly adjudicate a dispute.) I heard a case of a rabbi who was expected to routinely have massive meals for guests and asked if there was a financial allotment for that.
The only responsum that comes to mind off the top of my head is a school that demands its teachers attend the annual banquet (and pay a full-price ticket, which means a donation to the school). Rav Moshe Feinstein wrote that if explicitly stipulated as such in the contract, fine; but otherwise, the teachers would have paid a few dollars for dinner at home, but the inflated price of the banquet? Either the school should pay them the difference, or it can't demand that they show up.
I think that unfortunately, Halacha focuses too much on the minor details of the Mishloach Manos instead of emphasizing the general idea of friendship brotherhood, happiness, mutual support and love.
Therefore, the regular Halachic considerations of Mishloach Manos (like who's obligated to send to whom) do not seemingly apply here, if the goal of the Rabbi is to support and maintain the community.
We do hold that the members of the community (כופים בני העיר), let alone the Rabbi, can force the other members to participate in maintaining and developing the community. Therefore if it is commonly accepted that this kind of activity (sending out Mishlochey Manos for all members) is beneficial for the community, all the expenses must come from all the members.