If you are really sold on keeping that cup what about a logo sticker instead of a sleeve?? Or maybe a rubber stamp that you could roll around the side of the cup?
What I've seen done very effectively is using an ink stamp to customize the standard cardboard sleeves (or the hot-melt adhesive LBP). Paired with a plain white paper cup this is nice custom branding on the cheap.
Out of curiosity, why are you so sold on this cup and so against paper, given that this would be the logical choice for your custom branding strategy?
Hopefully someone else has something like you're looking for, though I suspect that many options would be cost-prohibitive. Good luck.
Our small chain of stores use a Festiva Solo cup printed with our logo on it. Our supplier warehouses them for us but we have to pay for them first. It can be a problem if you have 4 sizes. They are a foam insulated cup that retain heat very well. We would like to go to a paper cup but I find paper to hot to hold and sleeves are another added expense and more work. We try to encourage our customers to use our travel mugs by providing discounts on drinks when they use them.