If you don't have a wedding website (and I don't think they're OTT, but if you don't want one, that's cool), you can just tell people what you want when they ask you. And they will ask you (or a close relative), believe me.
I didn't have a registry, and we didn't put any gift info on our wedding website. Guess what? We still got gifts! Some were tangible, some were cash, some were charitable donations (if anybody asked, we told them we wanted them to donate to a charity of their choice in our name).
So your fiance is worried that 'word of mouth' won't travel far enough. To that, I have to answer 'so what?' Is your fiance seriously going to be put out if people don't buy from the honeymoon fund? People aren't obligated to get you the gift you specify. In addition to that, it's been my experience that in the absence of a registry, most people give cash, which you can put towards your honeymoon.
Please, don't put in any business cards, or cryptic 'phone for info' messages. For one thing, I'd assume that was for help with finding hotels or something. Either put the info on your website (not on the front page, but I think deliberately clicking a link on the home page counts as 'asking for the information'), or let word of mouth travel.