So I don't know what the answer is, but I don't think it's "just" removing the "costly signal".
For a long time, in fact, this mindset meant that a really effective way to "hack" the Congressional contact system was to send your message in a telegram -- because if you were willing to pay to have a messenger run your message across town and deliver it by hand, you were assumed to be very motivated indeed!
(I doubt that's true anymore, of course; not least because I'm not sure you can send telegrams anymore. But now I'm wondering if you could get the same effect these days by sending your message via FedEx's most expensive service...)
That being said, once the sums you contribute get large enough you will almost certainly get to be known around the office -- either explicitly (e.g. the Chief of Staff tells all the people who handle correspondence "Bob Smith is a good friend of the Congressman's, if you find any messages from him bring those straight to me") or implicitly (e.g. the staff of the Congressional office notices that when Bob Smith comes in to meet with the boss he's ushered right in and stays for a long time). So large contributors' messages can easily get special attention, even without that kind of direct connection to their contributions ever being made.