Take a minute and read a few of those cheap bids. Do they inspire confidence? If you were an employer, would you honestly believe that the person who wrote that bid is capable of building the thing you're trying to build? Of course not. They all sound like a bunch of desperate children trying to get away with something. If you want to take work from them, all you need to do is not sound like a desperate child trying to get away with something.
Take 10 minutes and write a good proposal, with a summary of the project, your basic approach to solving it, and what you think it would realistically take to do the job. Quote your full rate, and don't worry even for a second that your bid is ten times higher than the next highest one. You're sending a message that "I can actually pull this off", and the best way to do that is to distance yourself as far as possible from the herd.
If you succeed, the project owner will end up looking at two stacks of bids. One stack will have 150 flaky looking quotes to do the whole project for $300, none of which stand out as inspiring much confidence. The other stack will have a single well written proposal, quoting a bit more than he'd expected to pay, but clearly from a guy who has done this before and can do it again.
His choice is now: Sift through that rubbish pile and hope I get lucky, or go with the expensive guy.
That's a pretty good place to be.