I know this isn't the answer you're hoping for, but since you do mention it toward the bottom of your question, I think it's still on-topic; yes, you would (probably) be more sustainable to just buy organic tomatoes at the market.
Inefficiency of Do-It-Yourself (DIY)
This is a recurring theme I see a lot: people mistaking sustainability for DIY. The two are not the same.
One reason is kind of a corollary to an argument that's common in economics. The argument is that an economy works better when each of us specialize in what we're good at, and leave the other work for people with different talents. If you're really great at growing tomatoes, I suggest you take up tomato farming. If you're better at something else, do that for a living. In economics, you do have different priorities (vs. sustainability). However, they're not completely unrelated. Economic efficiency focuses on costs. Sustainability focuses on energy, pollution and finite resources. But energy and resources cost money, too.
The organic tomato farmer is also trying to run a successful business. They don't want to use more energy, water, organic fertilizer, or land than they need to. They'll try to minimize those things, and given their experience, will likely do a better job than you do. Economies of scale matter. It's hard to have economies of scale in a backyard garden. Do it if you love to garden, or want the freshest possible food. But, don't assume it's most sustainable. You should plant the kind of plants in your yard that are native to your area, or provide habitat for birds, insects or animals we do need more of.
Here's also a link about the relative importance of "food miles" in the energy equation, which I think tends to be overrated. For example, if transportation to market represents only 4% of greenhouse gas emissions for a certain food, are you sure your home farming won't be more than 4% less efficient than the professional farmer?
There also appear to be a few assumptions in the question that I'd challenge.
Using a cat or dog would be sustainable (if not for allergies, or lazy cat syndrome). If a dog or cat is active enough to be any good at chasing squirrels (and probably only when they're young), then they'll need a high-energy diet. Such a pet, pound-for-pound, is going to consume much more food energy than a human. Food takes energy, land, water, and soil to grow. Pets can have a large negative environmental impact, even though they're "natural". Keeping one just to protect your tomatoes would be like buying a schoolbus to drive to the grocery store, but claiming it's sustainable because it's a hybrid.
Shooting squirrels would not be sustainable. I vehemently defend species under threat from human activity, but not all animals fall into this category. Mankind has generally chased away the squirrel's natural predators, so in many areas people live, squirrel populations have increased. Thus, killing squirrels isn't necessarily unsustainable (especially if you, or local birds, eat them). Yes, squirrels do help propagate trees, but where you live, there's a good chance that humans decide which trees get planted, and which are removed. If humans perform this function, we can probably tolerate a reduction in the number of squirrels doing so as well. For balance, I'll link to PETA's defense of squirrels.
Growing your own peppers would be better. This is really just about the specialization argument again, and how doing everything yourself isn't the point of sustainability, and usually isn't efficient. The relative sustainability gains you might achieve in growing your own peppers are likely small compared to the losses of growing crops that are eaten by overpopulated rodents. Worry about the tomato problem first.
If you really need a narrow answer to this question ...
You might need to provide more information. Why are tomato cages not practical for you? We may need to understand this is in order to give you another recommendation. Cages would have been my suggestion, and in general, I think the MNN article covers most good alternatives.