Here is a general overview of how one can conjure and seat a spirit of the dead. While I have never seated an animal spirit into a vessel, I don't see why it couldn't work.
First, you want to obtain the vessel itself. If you are a skilled artisan with clay or wood or other such materials you can certainly make it or you can purchase it. Doesn't have to be anything specific but common vessels are jars, small boxes like jewelry boxes, toy/doll coffins, hollow-clay dolls, etc. Once you've obtained the vessel you would then want to cleanse and bless/consecrate it for the purpose of housing a spirit of the dead. In this instance, I would suggest doing so through the office of a spirit sympathetic to necromancy, such as St Cyprian, Bune, any of the various deities associated with death or the dead or any spirits you personally venerate that govern the dead. Once the vessel has been consecrated for use, you then want to begin gathering items that will go into the vessel in order to attract, bind/seat and constrain (doesn't have to have a negative connotation) the spirit to that vessel. These items should not be random items and should be sympathetic to necromantic magic, death energy/essence and the type of spirit you're seating (what attracts a Slyph Elemental may not be the same thing that attracts a shade of the dead). First would be the obvious items like roots, herbs, minerals, bones and a few crystals to anchor the spirit and also amplify the energies of the other items in the vessel. Some examples of necromantic items would be cypress, mandrake, graveyard dirt, used and rusted coffin nails, human hair or fingernails, human bones (usually a skull but can be any bone), and depending on the kinds of tasks the spirit would perform other items would be included as well such as bullets, a knife or machete, animal organs like hearts or eye balls, etc. But this is for an animal spirit so I'd say use the above examples as inspiration for what could be substituted to better fit an animal spirit. However, you definitely want to put at least one of the bones from the animal's skeleton (I believe you said you have it) into the vessel. Then you'd want to consecrate the items for the purpose of housing a spirit of the dead. The vessel and the items can be consecrated at the same time if you'd like.
Once you have your consecrated vessel and items, next you'll want to seat the spirit. Since this is an animal spirit and you not only have it's bones, but a personal relationship with the animal, conjuring them shouldn't be too difficult. You can design the ritual actions yourself but again I would suggest going through the office of a spirit who actively works with the dead, be it a psychopomp, a death deity, your ancestors, etc. It should be stated here that I have never worked with animal spirits in this capacity so I cannot say how the actual process of seating it will go. With humans, you either force them or negotiate a "contract" with them. I cannot say if it is the same for animals but I can't imagine it being much different. The ritual here would serve two purposes:
1) conjure the spirit
2)seat the spirit
Once you've conjured it you would then work to seat it. Its your ritual so I won't tell you how to seat them but you will have to convince (or force) the spirit into the vessel. Since this is your pet, I don't see that being that big of an issue. Once seated, you want to thank the presiding necromantic spirits you utilized during the consecration and conjuring/seating rites by giving an offering of tobacco smoke or rum. Both must come from your mouth, meaning you'll have to blow smoke or spit/spray rum out of your mouth onto their sigil or effigy or what have you. If you don't smoke, that's fine as you don't have to inhale. Same if you're uncomfortable with drinking the rum. If you're still against it I would seek other appropriate necromantic offerings such as coins, blood (preferably not yours and an animals) or incense smoke. After thanksgiving has been done, you now have a seated spirit within the vessel. You will want to feed this spirit periodically. Usually fetishes are fed blood, rum, tobacco smoke, etc. Good luck. Cheers mate.