Interesting question! Some thoughts:
1. I wouldn't say there is a conflict, but it is clear that not all economists are happy to accept behavioral forces at all or as being equally important. The benchmark of agents behavior is still frictionless optimization of a standard utility function. True, in many settings behavioral forces are not very important, but even when they are clearly at play, other explanations are preferred by most economists. Even if these explanations are more convoluted, as long as they maintain the standard assumptions. I recommend reading Chetty's Ely lecture:
http://www.rajchetty.com/chettyfiles/behavioral_ely.pdf
2. It is definitely not incorporated into the mainstream. Most models and empirical tests will not take into account self-control or reference-dependence or any of those.
3. Even within behavioral economics there are several schools of thought -- roughly the biases and heuristics vs bounded rationality -- which of these is "mainstream"? unclear.
4. Some fields are more open - most notably I'd say public, household finance, and development. Others much less so - probably macro and IO are the least accepting.
5. Today I would say that behavioral economics is still a field by itself. By analogy to development economics:
Development has some unique questions it tries to address (e.g. poverty and its effects), but any study on whatever topic in a developing country is also considered development. Likewise behavioral - core questions are what are the right models of economic agents (e.g. limited attention or reference dependence), but any study that incorporates some less-than-perfect decision making can be considered behavioral economics.
The first handbook of behavioral economics is in the press - gives a good idea of what's happening in any of the fields (see link below, but most chapters are available in their authors' webpages):
https://www.elsevier.com/books/handbook-of-behavioral-economics-foundations-and-applications-1/bernheim/978-0-444-63374-3
