@warrior-princess, I like to take pictures of birds too! I have found that bird photography isn’t necessarily one of those genre’s of photography that needs to be done at the edge of light. In fact, bird photos generally benefit from direct sunlight on the bird because it provides catch lights in the eyes and causes iridescent colors in feathers to pop. It’s still advantageous to photograph birds in the morning and late afternoon when the light is more colorful than midday, but I wouldn’t be too concerned about the aperture needing to be ultra wide. The fact that you are zoom in 200mm or more will also make the background very blurry. You don’t have to rely as much on aperture to make the background blurry like you do with wide to mid-range lenses. Like @dfrazer said, if you’re on a tight budget, you’ll probably have to rely on image stabilizer instead of wide apertures to deal with those slower shutter speeds.
Having said that, for bird photography, you really need more than 200mm. I use a 2x extender on my 200mm lens (which limits the aperture from f2.8 to f/4), so that brings it up to 400mm. If I put that lens on a cropped sensor camera, then it’s the equivalent of a whopping 640mm. Though I still wish I could get closer sometimes, and though my bird photos aren’t incredibly sharp, it works well for me.
So, you really need 200mm at the minimum. And I don’t think you are going to find a telephoto lens of that focal length for $200 or less, especially with image stabilizer (let alone, an f/2.8, or even an f/4 aperture). You will really need to save up more. Find your ideal lens/setup, and then start saving toward that goal. It took me many years to get to the setup I have right now.
But, that doesn’t mean you can’t do close-up bird photos with 200mm or less! Until you can get a better system, bring the birds closer to you with a bird feeder and shoot from a “blind” like your living room window!