Been there, done that! My daughter was 9 when she first started asking for a dog. I grew up with dogs and love them, but my husband does not. When we met and married we were both cat people -- I also love cats, and a cat is a good pet for people who are always on the go because they are more independent and low-maintenance. You can leave a cat by itself for the weekend as long as you put out plenty of food and water and have a litter box, but you certainly can't do that with a dog!
So anyway, when my daughter first starting asking for a dog we said no because of our crazy schedules. For two years she kept asking, and when she was 11 I felt that we were at a better place and could give a good home to a dog. So I started supporting her request and my husband gave the same answers as yours -- no, and if you get one I'm leaving. They are messy and require too much time and attention, he said. Now, I'm a firm believer that both parents have to be on board with such a decision, otherwise it becomes a point of contention if one gets their way over the other. But I began resenting the fact that he was essentially getting his way, so my response started being "we'll miss you when you leave, but at least we'll have the dog for company." At times I even considered just bringing a dog home despite my husband's protests (but knew in the end that would be a bad idea!). Finally, I pointed out to my husband that our daughter is, quite frankly, an OUTSTANDING daughter (excellent grades, athletic, mature, helps out with her little sister all the time, etc.) and this is the one thing she really wants. Well, he couldn't argue with that logic and he finally relented and my daughter was presented with a puppy that Christmas, she was thrilled!
We made it very clear to our daughter that she is responsible for the dog -- feeding, scooping the poop, walking, baths, etc. Of course, she didn't fully comprehend how much work that would really be (sort of like first -time parents!). And it's not always easy because sometimes she's got a lot of homework or is off at practice for her various sports. But my husband, for all the fact that he's still not a dog person, has been willing to help out with walking the dog and playing with her sometimes. He is overall good to our dog. And of course I help out as well, and I'm overall enjoying having a dog again.
But even a well-trained dog can be messy! Our dog is fairly well trained (took training classes, etc.) but she still acts up when she's bored (like any chiild!), with one of her worst offenses being breaking our fence (this one really ticks off my husband because it creates more work for him). She literally breaks through the fence to get out of the back yard, although she doesn't run away, just sits at the front door waiting to be let in (at which point my husband wishes she would run away!). And both my daughter and I have learned to dread rain, which usually means muddy paws. Our dog has managed to wear out the grass in a couple of spots which creates mud bogs when it rains. So we can't let her into the house without first doing a thorough paw cleaning, and sometimes we don't do a good job (even though we thought we did) and then mud gets all over my kitchen floor and rugs, ugh! And after a particularly hard rain, it means we'll be doing this for a week and it gets very tedious and I'm constantly spot-cleaning my rugs. I've learned to appreciate drought conditions, LOL.
And even after 1 1/2 years, our cat is still not a happy camper! The dog and cat do not get along and our dog frequently harasses the cat. Although sometimes I get a kick out of it because when the cat gets fed up she will go after the dog and the dog runs, it's so funny! But this, too, can also get on our nerves and we can't seem to train this bad habit out of our dog.
So, all I can suggest in patience. I'm assuming that, just like my husband, your husband's the threat of leaving is an idle threat and is just his way of expressing just how much he doesn't want a dog. Never once did I believe my husband would actually leave! But you sound worried that your husband would actually leave, which may speak to a bigger problem.