Ever since before we got married, dh has wanted a dog, but I was unsure for a lot of reasons - I don't like the smell of dogs, found the thought of picking up their poo somewhat nauseating, and wasn't sure I was up for the responsibility (as a SAHM, I'd be the one doing walks etc during the day). Plus we'd got cats, and I was really worried that the cats would hate the arrival of a dog, and be really miserable, and as they were there first, that seemed really mean to me.
Luckily for me, we lived in towns where a big dog (dh always wanted a labrador) wouldn't have been very happy, and dh's commute meant that he'd see almost nothing of a dog, so it remained a pipedream for him.
Just over a year ago, we moved to Scotland, and our surroundings are far more dog-friendly, and where dh's commute is much more reasonable, so the dog question came up again, but this time, having spent time with our neighbour's dog, and the puppy next door but one, I had started to come round to the idea.
It has taken a year, but I have got to the place where I am happy to give dog-ownership a go, and a fortnight ago, a brown lab puppy came into our lives.
And my worries? Well, picking up the poo isn't my favourite job, but doesn't bother me, the cat seems to be ok with the puppy (not happy yet, but hasn't left home and doesn't seem miserable), and I am enjoying the puppy's company so the responsibility that she will be, once the children are back at school, seems fine and I think I'll enjoy it. I still don't like the doggy smell, but even that's not such an issue as I thought it would be.
What I am saying, in all of that rambling, is that it might be worth waiting and giving your dh time to come round to the idea in his own time. Perhaps you could also 'borrow' a dog for a while, or look after one whilst a friend is on holiday, so that your dh can see for himself that there are positive benefits to dog ownership without you having to make a permanent commitment to a dog. He might see that as a reasonable compromise.