Thats a good question, I have done it twice. First time I was in cargos/white tee.. Second time, I was comming from a meeting so I was looking upscale with button-ups, dress pants and blazer... But I have never really thought about tht

I used to worry about this since I am pretty much a t-shirt and jeans guy myself. Now I figure as long as I am clean, respectful, and bring the correct donation they really shouldn't care what I am wearing.

and only see the "finest gentlemen", and they claim that their clients aren't like the typical client and so on and so on.

Well guess what, I emailed them and said I'd like to see them. They said "sure!". I showed up, dropped the envelope, had some fun, and left. Nowhere along the way did they ask if I was an upscale gentleman of the finest caliber who was worthy of seeing them.

I normally wear a t-shirt and jeans, but when I go see a girl I make a bit of an effort and put on a polo shirt. If I dressed like I was going on a date my wife would wonder what was up.

"Hey, that's me. I'm an upcale, distinguised gentleman!"

This just means she wants you to pay well without complaint.  If you pass screening, I can't imagine being turned away because I wasn't dressed well enough.  I do dress as I would for a casual date.  Most often a polo shirt.  If a button down is in character for you then, wear it.  
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Seriously, if you act like a gentleman....

If you act like a gentleman, you have good hygiene and of course you have the correct consideration....they could care less what you're wearing, as long as you look presentable.

Added note: If your going out to dinner, etc. make sure that you dress appropriately. I'll never forget a dinner date I once had. I walked in looking like I was going to a fine dining establishment, which we were. The gentleman, on the other hand, had on gym shorts and a ripped up t-shirt and was smelling ummmm ripe! I asked him if he brought along some clothing for dinner and he said, we could go through his suitcase to see what we could find. He proceeded to place his suitcase on the bed. When he opened it, it looked like the bottom of my kids closet...with clothing waded up. Come on...really? Needless to say, I was totally embarrassed during dinner and felt like crawling under the table...and not to service him either.