This suggestion may never see the light of day, but I believe it is a true win/win/win.
Kill a selected number. Have local butcher volunteer cutting and wrapping. Donate bulk to the soup kitchen. Donate remainder to food bank. Have soup kitchen prepare special venison dinner fundraiser evening; have local wineries involved; price public tickets reasonably — I would pay $40 for a couples ticket.
The hungry will benefit from the protein = win. The needy will get fresh meat = win. The public will increase soup kitchen coffers and develop a deeper understanding of how truly great their community service is = win.
Repeat this process with geese. Do both whenever required.