Anywhere in the Lakes is always going to have second homes, self catering and lots of B&B, due to the amount of tourists who visit the area. In excess of 20 million per year, so for all those who dislike the visitors and offcomers, many locals also understand the financial benefit to the area, so it is very much a balance.
Many locals run tourist businesses and have sold houses to the second home owners at high prices, whilst then complaining that locals cannot afford property!
The ages of your children, there are some lovely well supported local schools, Lorton being one of them. Keswick and Cockermouth of course have senior schools.
Becoming part of the local community is possible, there is always plenty going on in the small villages plus in the larger towns too, what locals do not like is people who come in and then want to change things, from little changes to local events through to making huge changes to houses. Move, get to know the locals by offering help, attending events, volunteering etc but do it in a calm, quiet way. What you will find is over time you will begin to fit in and then for example if you wish to build an extension onto your home, you are more likely to get support from the locals with the planning process.
In any area there will always be people who seem stand offish and not accepting of others, this happens as much in Cumbria as other areas. Being a large farming area, you have to understand many of the land owners become a little peeved with visitors and offcomers who let their dogs chase their sheep, try and tell hill farmers that the area should be rewilded and everyone should be vegan! It is not a good way to gain the support of the community! My parents farm in Cumbria, they are lovely (of course I would say that!) but my father in particular can have a whole conversation, leaning on his shepherds crook, and not saying many more words than "alright" and "aye"! That is how many Cumbrians are - we are not rude, just sometimes we don't say much but we are also a very welcoming and friendly bunch of people.
What I suppose I am trying to say is everyone will have a story of so and so large landowner was peed off because so and so purchased a house which they wanted so and so to buy. It happens everywhere. When moving to an area and buying a property you have to do your own investigation and make up your own mind as to if it is right for you.
Keswick and Cockermouth are very different, the obvious one being Keswick is a major tourist area but it is in the centre of some of the best fells and outdoor areas the Lakes has to offer. Cockermouth has more of a market town feel. You mention Lorton and Bassenthwaite, of course they are much smaller so difficult to compare with Keswick and Cockermouth.
If you want land, then you will obviously be more rural to get that, there are many lovely little hamlets and villages near both Keswick and Cockermouth, I think your biggest difficulty will be finding a property for sale, irrespective of how big your budget is!