Well, at least you didnt say, and our cat, Chloe."
Id say it all depends on the big dog. The little one doesnt take up much space, although you have to take care not to bait your hook with it in the heat of the moment. But if the lab can be persuaded to sit still, then you should be OK.
What you want to buy is a tandem sit-on-top kayak. Boats such as that are easy to manage, work well when fishing, and have the space to accommodate human passengers plus a pet or two. Take a look, for instance, at the Ocean Kayak Malibu Two XL ($800). Its a large, stable boat thats designed for paddling around quiet water, which seems to be what you have in mind. Where the pooches sit is a bit of an open question, but there should be room for each to sit between your knees. The big dog may also be able to sit in the center seat.
Wilderness Systems Tarpon 130T ($800) has a very similar layout. You might try to look at both, sit in each, and imagine what it would be like with two canines. Also shop around for a NuCanoe, which is designed to be ultra-stable when fishing and is a nice mix between a sport canoe and sea kayak. They start at $799.
Given the dogs and the fishing poles and so on, there is something to be said for getting a boat that you paddle with your feet, like a bicycle. Such a boat is the Hobie Mirage Oasis, which has two sets of pedals for operating its flipper-like propulsion system. You also can use oars, of course. The Mirage Oasis is a little longer than the other two boats Ive mentioned here, so can carry a bit more. And its easy to add rod holders and other fishing-specific accessories. Down side is the price at $2,500.
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