I have a cottage that I offer for rent, but we don't take pets, so I can't offer it to Rhode Islander. But I can say that HomeAway and VRBO are very user friendly.
I put "Lake Winnipesaukee" into the HomeAway search box using the random dates of August 6 to the 11 and "4 guests." You can then use other filters, so I clicked on "pets considered." The original search turned up 62 responses; after I filtered for pets and waterfront it came back with 6 responses.
Here's an example:
https://www.homeaway.com/vacation-rental/p127751vb
I'm sure you'll find something suitable!