Sorry, this is more a brain dump than a structured post.
I did, for a number of years, exactly what you describe. I was very clear that I would only take photos of dogs in natural surroundings, doing what the dog wants to do. I set out not to capture perfectly posed photos, but photos where the dog's character will shine through.
So, I worked primarily outdoors with the occasional indoor shoot (NOT studio). And you know what, I got work all year round and people mainly wanted the outdoor stuff. People did not complain about their dogs getting dirty or wet, or the cold/bad weather. This is stuff they do with their dogs on a regular basis so it was just another day walking the dog for them, just with a photographer, and in a different place.
Now, I started out by offering my time to a rescue centre. I did a number of outdoor shoots in their paddock which honed my skills somewhat. My first paid shoots were entirely to benefit the rescue and even when I was making more of a business out of it, I still made sure the rescue got a donation from each shoot. This helped the rescue out greatly, and also provided me with some marketing.
Woods, they are great in the summer, but as winter draws in not so. Woods are dark, so shoot right at the edge of them in Winter, and make sure you have a handy field nearby too. Remember, if you are shooting for clients you really should get permission from the landowner too, even public woods may have some clause about commercial photography.
So, the shoot itself. You need to be 100% confident in your gear and your personal style. You also need to put the work in with your client before you go out and shoot - get to know them, and more importantly their dog. You must be entirely comfortable with both otherwise this will quite possibly show in the shots. Be prepared to get mucky, and only get your camera out if you can recognise it as a tool that is going to have to take some abuse. Dogs are playful, playful dogs, much and cameras rarely has more than one outcome... Make sure you have a backup camera and lens that you know how to use. Your client may have travelled a good distance for the shoot, equipment failure is not an option.
What you want to do is possible, but with limitations. I loved doing it because the people that would travel 100+ miles to me for my natural dog shoot were, without exception, the best sort of dog lovers I have met. Their dogs were more important to them than anything and this showed through. I do still miss these shoots and once time permits I will be starting up again.
Might add some more thoughts in a bit.