My answer is that it would be unequivocally immoral in this case.
The life of a dog, let alone one you adopted, has moral worth rendering moral obligations towards it. It's not just a means to an end, but has its own moral value.
Indeed a God inspired moral outlook would claim that the dog is not merely a means to some greater to good, but to an extent an end in of itself. And therefore a wrong action towards it can't be justified by any kind of calculation of benefit, except in very extreme cases perhaps (in such extreme cases, such an action, while harming the dog of course, in being justified, wouldn't be a moral wrong).
Admittedly in this mode of thinking I've elevated the dog's moral status.
It might be the case that a dog's moral worth is lesser, and it is not a wrong to kill it in the interests of humans in a broader set of cases.
So you have to ask yourself: to what extent a dog, let alone one you adopted, has a moral worth, making him an end in its own right? And even if you permit that as an animal his life can be forfeit in the interests of humans, in some cases, so cost-benefit analysis can have greater validity, in principle, you still have to ask yourself is this remotely such a case (and does it make a difference that it's a dog that you adopted and trusts you)?
It sounds like a good solution would be to find a new home for it (perhaps you can even offer to pay the new owners $200 for the first four months).
In my view, you can give it to a new family, but to kill it would be highly immoral in those circumstances even if the dog is assigned no more moral worth than a person with the average moral intuition would assign it.
I later saw that you said you love the dog. So no need to even give it away. Again if the dog has no moral worth in of itself, like an inanimate object, you can trade it freely. Once a moral worth is assigned to him, it may the kind persons are typically assigned with, making cost-benefit calculations nearly always wrong. Or it might be assigned a more modest worth as typically assigned to home animals, which more freely justifies profitable trade off at the cost of harming the animal. But even in the latter case, nearly everyone assigns a home dog a greater value than would make permissible the killing of a home dog except in pretty extraordinary circumstances. I'd go with the moral sense of the majority on this one.
PPS - response to a comment:
"This question appears to argue that a dog's life has intrinsic value conferred to it by God. However, from my limited familiarity with the Bible: There is indeed a commandment "Thou shalt not kill", but it is usually understood to mean "kill humans". My understanding of Christianity is that killing animals is not necessarily a sin."
"intrinsic value conferred to it by God" - I would have said that God anchors that moral truth in the fabric of reality, and it emanates outward like an electric field, rather than gives directly.
Here I used God as anchor of moral value.
If you don't accept that at all, then you have to ask yourself, does a dog have moral value anyway? And if yes, is it the elevated kind or the kind more commonly associated with animals (historically)? If you answer that it has no moral value, or such moral value that permits killing it readily, I'd say that you're going against the moral intuition of most people, so perhaps you might want to look further.
If you accept God as an anchor of moral value, I think any theistic believer would argue that it follows that at a minimum animals have to be treated with kindness and respect as they do have some moral value. It might not be the elevated moral value associated with persons, but even on the lower rung, it is sufficient to prevent the killing of an animal under the circumstances described (especially in light of the trust bond between an owner and any pet, let alone a dog).
And specifically to Christianity, while I agree with you that killing animals even in cost-benefit circumstances is understood to be permitted (even the killing of humans is sometimes understood to be permitted and justified too), it depends greatly on the circumstances. In these circumstances, of first taking an animal to your trust as member of the household, and then "volunteering" it to be killed, while presumably you don't significantly deprive yourself in the process, is something that's sooner in the spirit of Mammon and such than in spirit of Christ.