Some suggestions:
This tip was passed along to us by a woman who lived in a Catskill Farmhouse for years, and even though we were suburb dwellers, it turned out to be the only thing that helped get rid of our mice. Take plaster of Paris (dry) and mix it with chocolate milk powder and set it so the mice can eat it. It will be a dry powder. The good idea about this mixture is that the mice will leave the house to find water after they eat it so they will not die in the house. It is also basically non-toxic, having no poison in it. It worked for us, and we had quite a few mice that we could not get rid of other ways. I guess mice love chocolate too.
Instant Potatoes
I have a rather frugal option for getting rid of field mice that an old farmer told me about and it really does work: Use instant mashed potatoes; they are much cheaper and alot safer than rat posions. Place a few tablespoons where the mice are seen most often; try to keep them out of reach of the dog, just to be safe. The mice will eat the instant potatoes and die because the potatoes will expand in their stomachs before they can fully digest them. Most likely, the mice will die in the house if that is where they are living, but it's better to have dead mice than live ones! As for the dog and his/her intolerance of a cat: you may be very surprised! We now have 5 cats (1 indoor, 4 barn cats) and I would have never thought my dog would tolerate cats. She actually enjoys them and plays with them for hours. Since getting the cats, my mouse problem is over!
Bacon Fat High Wire Act
Years ago, our weekend cabin in the woods was overrun by field mice. My father used an old galvanized water tub (2 to 3 feet across, depth doesn't realy matter), and stretched a wire across it, securing the wire to the handles of the tub. Strung on this wire was a tin can with a hole punched in each end so that it would spin on the wire. Coat the can with bacon fat and fill the tub with water.
The mice would go crazy trying to get the bacon fat - of course if they made it across the wire to the can, the can would spin them into the water and they would drown. After a week away from the cabin, we would often find dozens of mice in the tub.
Try peppermint oil. Peppermint oil is a natural deterrent. The smell is simply too intense for rodents and they will not try to go near it. It also helps to mask the scent of any tasty morsels that have been missed when cleaning. You can get peppermint oil in most health food stores and even some major grocery stores. Place a drop or two on a cotton ball. Place the cotton balls in areas where mice are likely to enter your house, by doorways or heat vents, etc. Another helpful deterrent is to grow peppermint plants near the entryways. You can use the mint in cooking as well as it serving a deterrent purpose.