Actually, a nengajyo can imply a quid pro quo business relationship or a continuing friendship. Also they might feel obligated to send you one next year. But since you gave them a special one, then they wouldn't want to send you just their normal one that they use for advertising to customers, thus meaning they must make a special one for you. Instead, I would suggest a gift of something from your hometown. Food would be great. Not more than 3000 yen or so in value. And, give it as you leave. You could attach a card to the gift with a nice message that they read later, but a normal card used for such purposes.

I've lived in Japan for about 25 years and have two of my own businesses. I've made lots of mistakes in these realms. None that were terrible and I'm always easily forgiven because of my nationality. Nevertheless, I've put people into awkward positions when my culture value of showing friendliness (for example) clashes with their culture value of providing a service.