Do your homework on the high end pellet guns.  Some can equal a .22 subsonic round.  Problem is, the pellet is light, and does not have good downrange velocity or energy.  Close shots will work well.  
Don't know how anyone can not have a .22 in the house.  Subsonic .22's (often sold as target rounds), can be found with hollow points and 40 grain bullets.  That would be my choice (CCI subsonic HP).  Even in the most liberal anti-gun states, a .22 can be had with little issue.  Get a good quality bolt action rifle (does not have to be expensive, and can be purchased used), scope it up and have fun learning to shoot.  Yes, you can have fun with guns.
If looking to kill a pest, shot placement is everything.  Don't care if you are shooting a .30-06 or a .22 short.  Learn where to hit your target to put it down quickly and humanely.