As long as the logo doesn't contain any visible trademarks; I can't see why a firm wouldn't do a custom t-shirt with the logo on. Just don't give them any ideas as to what it is and all should be good. There wouldn't be any repercussions to you or the business if a few t-shirts are printed for yourself and I don't think any awkward questions will be raised either.
It's mainly sellers on eBay that print 1,000+ of Call of Duty t-shirts and other valuable protected brands that cause a stir. I remember seeing something on the news ages ago in the UK when a small company was doing videogame memorabilia clothing and they sold tens of thousands of t-shirts for £12 or so with trademarked brands on Fruit of the Loom blank t-shirts and they made a fair few hundred thousand out of it. That's the shit they're clamping down on but any independent shop or online business who you send the logo to may just do it without raising an eyebrow as long as you don't give them the full explanation of the score