I was going to suggest #110s also, find runs and set up underwater run/den sets to keep them out of sight, avoid ducks, etc.
Ask for use of a cart. It will help you carry the load, will be easy on the gentle turf, will get you pond to pond much faster, and will help you keep a low profile by blending in with your cart easier. I would also suggest just keeping your appearance neat, clean, professional. Hip waders...put them on at the pond, not at the clubhouse or parking lot, and if it is warm out wear a polo shirt with a collar. That's what I did anyway. The course I took care of was actually one of my toughest jobs, but we each benefitted from each other. I learned how to stay low, professional, and dealt with their challenges posed, and they got a rat-free course.
I was allowed to trap during season, but then had to walk in only after the frost, and could never walk across any greens or even close to the greens in the rough areas. Had to stay on the little paved roadways as much as possible. after a frost, even footprints across the rough tended to turn blackish if it was cold enough. on the greens that would be permanent and they would never want a permanent pair or two of footprints across them. It often meant walking clear out of my way, carrying some equipment and catch, but it was the option we decided on. They either paid me well to do it in summer/early fall (when prices were worthless on the hides) and then I had to deal with golfers also...or...I was allowed to come in during trapping season, avoid a lot of golfers, but then had to be extra cautious of the turf and no cart.
Each course will have its different ownership, different challenges, different ideas on how they want you in there. Be flexible...if you can work with them on this, it just helps the overall trapping image that much more. On the other hand, it may also be opportunity for you to try out a new set or two and may find that this is the only place it will work, too. Good luck!
Jim