Rabbi Howard Jachter, in an excellent article on Halachic Perspectives on Pets (http://www.daat.ac.il/daat/english/halacha/jachter_1.htm), footnote 44, quotes "Rabbi Shmuel David (Sheilot Uteshuvot Meirosh Tzurim 38:6) (http://hebrewbooks.org/pdfpager.aspx?req=22624&st=&pgnum=270) [who] cogently argues that one who is accustomed to move his pets is analogous to someone who prepares a rock prior to Shabbat for use on Shabbat. In such cases the rock is no longer muktza since he has demonstrated that the rock has utility for him on Shabbat (ordinarily, rocks are muktza since they serve no purpose on Shabbat; once one demonstrated his use for a rock then it is no longer classified as muktza). Similarly, one who ordinarily moves his pets demonstrates thereby that they have utility on Shabbat and hence are not muktza". R' Jachter does not say whether he agrees with this statement or not.
So the answer according to Rabbi David is yes. I don't know if anyone else agrees with this.
Also, this quote does not clearly explain how to designate the rock: Is thought or speech enough, or do you have to do an action with the rock which demonstrates its use (e.g. using it as a paper weight, door-stopper, or a toy).