Can Ethical Vegetarians and Vegans Own Carnivorous Pets?

Introduction
One thing you learn pretty quickly in philosophy is that committing to a particular principle often commits you to unexpected positions. Since becoming vegetarian I'm starting to realize this a lot in terms of the principles that ground my vegetarianism.

Here's the issue: It seems that the principles that ground vegetarianism or veganism commit us to the conclusion that it's unethical to own carnivorous pets. Lets look at the principles that support vegetarianism and veganism and see how they apply to carnivorous pet ownership.

(Note: For economy I'm going to use vegetarianism to refer also to veganism and "pet" to refer to carnivorous pet).

Argument 1
The most common reason people become vegetarian has to do with Peter Singer's argument which , in abbreviated form, goes something like this:

(P1) Factory farming causes animals to suffer tremendously.
(P2) Suffering is bad and we shouldn't cause it.
(C)   We shouldn't eat factory farmed meat (or any meat where the animal suffered).

Notice this argument isn't against killing, only against suffering. The objection to killing a separate reason to reject eating meat which I'll set aside for now.

Here's the problem for people who accept this argument or something like it: Pets eat factory farmed meat. By owning a pet you are contributing to the suffering of many other animals--the very thing you oppose.

It seems as though having a pet is contrary to a vegetarian's ethical commitments.  A vegetarian can reply (in some cases): I adopted this pet and I wasn't going to just let it die.

Even if this is true, (P2) seems to commit us to a startling conclusion: we ought to euthanize our pets. Here's why. Lets take for granted that more suffering is worse than less suffering.  Your pet has had a pretty good life. Except for the time it spent before you rescued it, live's been pretty good. Almost no suffering. Now, keeping your pet alive over its natural lifetime causes the suffering of many other creatures equal in their capacity to suffer with your pet. Their suffering is of no lesser moral worth than that of your pets. If we are truly committed to (P2), that suffering is bad, and we should cause less rather than more suffering, it seems to follow that we ought to euthanize our happy pets. It will cause less over-all suffering.

Argument 2
If your vegetarianism is founded on the idea that eating meat is wrong because killing is wrong, there's an analogous problem.  Lets apply the famous trolly problem to see why.  A trolly is barreling down a track and will kill 5 workers working in a tunnel ahead. You can pull a lever and divert the trolly to another tunnel with only one worker. Should you allow the 5 to die or pull the lever, saving the 5 but causing only one worker to die? Most people say pull the lever.  It's better to save 5 lives for the "price" of one. So, if we accept this principle and say the lives of animals are of equal value (at least with respect to each other), it seems to follow that we should euthanize our pet (or at least not feed it) to save the 5 animals that will be slaughtered to feed our pet over its lifetime.

Argument 3
You can also run all the same arguments in regards to the environmental cost of raising animals for meat. Owning a pet causes more animals to be raised for meat. More animals=higher environmental cost=more bad. If you euthanize your pet, those future animals don't need to be raised for meat and there is less of an environmental impact. Fewer animals=lower environmental cost=less bad.

Weak Counter Argument: Causal Inefficacy
One way to reply is to say that "well, euthanizing my one pet isn't going to change the meat industry/how much meat is produced." Vegetarians hear (and reject) a version of this same argument when people object to vegetarianism. It's a bad argument. To see why, imagine the pre-emancipation slave owner saying "well, even if I free my slaves, everyone else is still going to have slaves and probably just take my freed slaves anyway." Is this a good argument? Nope. Owning slaves is morally bad regardless of what other people do. You're responsible for the moral consequences of your own actions; what other people do isn't relevant.

Conclusion
As a vegetarian, I don't like any of these conclusions but they seem to follow from the principles I've accepted.  At the very least, the principles suggest that, if we don't euthanize our rescued pets we at least have an obligation to ensure that all pets should neutered/spayed to prevent future pets from coming into the world. Also, that once our existing pet dies (naturally) we shouldn't get another pet--rescue or otherwise. That sucks. I can't imagine my life without doge.