Edit September 2022 - Sadly this place has put it's prices up by more than 25%, so I can no longer recommend it as a budget stay. It seems that it is because they have added a hot tub, so if that's your thing then maybe it's worth the extra.
After as long search I finally found a nice spot, in my price range, to stay in Franschoek. I had entered the 75km Maxirace trail ultra and decided rather to stay in Franschoek than drive out early on the morning of the race. After exhausting my normal search sites, I eventually found a spot on airbnb. I don't use airbnb, because I have had some bad experiences with being cancelled on at the last minute, and my bank charges a cross border fee when booking a place, even if it's within an hours drive from home. Luckily Franschoek Farm Cottages has a website, so I could book directly with them. They are not great at responding to email, so it is best to call. There are a few different cottages of various sizes, on 2 different farms. We stayed in Royal Gala, which is the smallest, and is attached to the farm house. There is no fence, it opens fully onto the orchards, so not suitable for runaway doggies. It is one bedroom, with a small en suite, with a shower, a open plan living area and well equipped little kitchen. There is a patio and braai outside. While I was not at the cottage much, spending the whole of the Saturday running, I can still say it was really comfy and quiet. Hubby and the boys went exploring through the orchards, in-between their crewing at aid stations, and enjoyed the mountain views between the trees.
We didn't get to do much exploring around the village, but did find a lovely dog friendly brewery and restaurant, Hey Joe Brewing Co. Their beers are excellent and the dogs even got a doggy beer to share, which was obviously delicious as they drank it to the last drop immediately. They have a large outside seating area and a huge lawn with a pond. So loads of space for the dogs to play.
A lot of the area that I ran through was quite easily accessible, being on dirt roads, or through farms. And the dogs were allowed at the aid stations along the route, to welcome me with fuzzy cuddles and sing me away when I ran off again. So while Mont Rochelle is managed by Cape Nature and therefore doesn't allow dogs, there seems to be a lot of areas were dogs could run.
So now that I have found this spot we will definitely return when we have more time to explore. Hopefully there will be some wine farms that allow lovely dogs.