I have to agree with everyone else here. You have no leverage, and you should just be satisfied with being able to find employment with the way things are.
Also, I don't think you really want to start your service with a company on this kind of note. Even if they bumped your salary, or gave you the signing bonus, they are always going to look at you as over paid, or greedy, or a combination of the two. Your future raises, bonuses, etc. would more than likely be effected due to this, and be lower than what you would have gotten otherwise. After a few years, the $2,000 extra you got at the beginning will be nothing compared to the 10 K you lost out on through the smaller pay increases.
I worked with a guy that had come to our office from his first employer. When he was offered the job, he had the mindset of, "I already have a job, if you want me to come and work for you, you need to make it worth it for me". They negotiated a higher salary, and he came to work with us. After our yearly reviews, they all took place at the same time, we would have discussions about how the increases were in terms of percentage. He got increases in the 2-3% range where the rest of us were getting 5-10% increases. I don't know what he was making, but I'm sure it wasn't much more than about 5K more than the rest of us with similar experience. At the end of the day, he got what he wanted in the beginning, but has paid for it ever sense. I'm sure by now he has been surpassed by those that he was making more than when he started. Be careful what you wish for.