I have a good friend who has a dog and my friend loves kayaking. I was thinking "why not bring your dog?" so I set out for a little research on the subject and decided to blog some while I was at it.
You can always get a Tandem kayak and have the dog in the front or buy a canoe but if you already have a kayak why not just put a small rug or chunk of carpet under your Bungee's on the front of your kayak? Looks like it will work and if you have a larger dog use a larger rug. I have kayaked with small dogs in the front seat of my tandem kayak, they were quite skiddish at first but they soon got in the kayak and enjoyed the view and the different smells of the outdoors, and seemed to adjust quite quickly.
The below was from WestCostPaddler site.
The above example they used an old towel but I am thinking a rug or a carpet sample would work quite well and would be more ridged for good grip. Snaps could be used to secure the rug on the kayak when in use too.This one (above) looks like the kayaker built a platform of sorts for his kayak. Source: http://www.westcoastpaddler.com/community/viewtopic.php?f=1&t=4798
WestCoastPaddler dot com has a great forum discussion on the subject, also check Paddling With A Camera too at http://www.fit2paddle.com/photokayaker/C1666358270/E20061006160351/index.html
Hope this helps and gives you some options.