Not sure if you need to limit feed - Here's what's worked for us, with 3 different cats, different times.
Moved to a different farm, the 2 cats we brought with were locked in barn for 7-10 days with run of the barn ( dairy barn with a T add on. They fed and watered there. Then we'd let them out daily and then lock them up at night. We'd call them in at night with a cat treat (store bought ones or cheese).  This worked very, very well. Never had cat's up at the house begging or trying to get in. When those 2 passed on, another stray showed up - same deal, put in barn (although, not locked up, hopping it run away). and it prefers to stay in there and will not come around the house or eat cheese or other human food.
They all are great mousers and always have unlimited store bought cat food. Which by the way, in the summer, hardly eat any and into fall/winter, they eat lots.

Cats are very social - but, independent . . . . the dog is setting a bad example also. They want to be around it. My cat, it has to find squirrels, mice and other varmints to socialize with . . . ..  I've been thinking of starting a Cat Facebook for it ? lol