I think the danger of any trip is probably relative to your experience. I have lived in the desert of the Texas Panhandle most of my life and what you guys call crossing the bay is open water in a hurricane to me. Some of the video I watched had rough water, rain, fog, and poor visibility that kept the people tied to a dock for several days on the east coast.
Doing the loop with small outboards and jet skis works when you're 20 but very few in their 70's would try it.
I think if you want a safe boat in a hurricane, you will need a heavy steel boat that barely floats, brought ashore and sheltered in a cave! ;-) (Cave to be well above sea level.)
My intuition says that if you have a solid deck between the hulls of a catamaran, things will get exciting, or at least uncomfortable, as soon as solid water starts hitting the deck, whether from above or below. I think that means you'd need a relatively large catamaran for rough seas, though I could be wrong. A Hobie 14 is pretty good in rough seas with a normal load, but that wouldn't be much good for you on your trip! That hull shape might be good if you were going large enough. I noticed that, at least in small chop, the ride was surprisingly smooth even when going moderately fast, perhaps due to the relatively small waterplane area, or whatever you call it. However, on a heavier cat, with wider, flatter hulls, I suspect the ride wouldn't be as smooth. Also, if you have a lot of initial stability, in large waves, I've heard the boat will move more quickly, possibly throwing you around. Some people really prefer the relatively level attitude compared to a monohull sailboat, but that doesn't seem relevant under power.
The following is predicated on Fast Fred's comment about speed restrictions.
If I really wanted to do a trip like this on a low budget, I'd probably look for a used twenty or 25 foot centerboard sailboat with reasonably shallow draft. If it's a popular model, the rig and sails might be sold to recoup some of the cost*. Around here, old sailboats that need some cleaning up sometimes go for ridiculously low prices. Like this one:
http://boston.craigslist.org/bmw/boa/5716009031.html
The centerboard is for shallow draft. Sailboats have displacement hulls, so at slow speeds they're likely to have a smaller wake and use little power. My impression is that they'll deal with larger seas than a typical motorboat. I remember being in a 30 foot sailboat on a very rough day on Lake Champlain. We were just fine, although steering took a bit of work. The boat wasn't rolling much because of the sails. We even stayed dry. The only other boat we saw on the water at the time was a houseboat of about the same length, which was slamming into the waves and covering itself with spray. BTW, in the summer, Lake Champlain is really nice most of the time. Great views of the Adirondacks, too.
If I was going to build something inexpensive for a trip like this, and I was worried about rough weather, I might choose a big plywood dory. Perhaps a St. Pierre dory. But I think that's overkill. Caveat: the only one I've ever seen in person was being built by a neighbor when I was 4. I saw it on the water, too, but never had a ride. In real life, I probably wouldn't build anything this big, and if I did it would have to be a decent sailboat too.
If I really had to have a catamaran, and I was concerned about rough weather, I might start with one of the Wharram designs, and leave off the rig. Some of them are like two anorexic dories fastened together, which I'd guess makes them relatively simple to build. However, I think you'd find the accommodations a bit spartan, and the deck smaller than you'd like unless you like walking on a net. If you made a solid deck and bigger accommodations, I think the seakeeping ability would be reduced. But again, I think a standard Wharram cat of reasonable size might be overkill. Caveat: the only Wharram cat I've seen in person was one on Lake Champlain that I caught up with using a Windsurfer.
Absent the speed restrictions and the desire to sail, I'd probably go with something similar to a Sneakeasy or a Tennessee.
*A reefed sail might be good for steadying the boat in rough water, but probably a shorter mast and smaller sail would do.