Well ,I did a google search on the smallest recommended rifle caliber for deer. Several authors noted that you should keep your shots at 150 yds or closer due to needing proper bullet placement with something so small of a caliber as wind can be a big factor with a little bullet. As several noted there are much better choices than a .20 caliber bullet for deer. The Arizona Cous deer link states that these are the smallest deer and that you should hunt with a .257 or bigger due to wind pushing little bullets.. Here are a couple of links. http://www.fishandhunttexas.com/debate_over_22_center.htm
http://www.arizonahunting.net/coues.htm
Will a 22.250 work? Yes. Butchers kill 1000 lb steers with a .22 rimfire to the head every day. Hunting is all about shot placement. Are therebetter choices to hunt deer with..I will let you decide? I have my opinion and that is what makes America great..freedom of speech and opinions.