What are the benefits and drawbacks of fitting one's problem within an advisor's research? I am passionate about working on a particular problem and domain, and have an opportunity to complete a PhD that does not require department/university subsidy. However, the problem I am interested in is outside of the domain of the research faculty. I have been encouraged to find a small part of my problem space that fits with the research of available advisors. However, I feel like conducting my research at another institution, where the research faculty are actively working in the same domain as myself, may be a better choice.

What are the benefits and drawbacks may be to work with advisors directly related to one's own area of research, rather than trying to just fit one's research into the scope of available advisors' research interests?

There is some discussion of this regarding the master's thesis, but I feel that the doctoral level is a little different, in that the advisor/advisee relationship is perhaps more important.
With regard to choosing to do your PhD research with a group working in your field of interest:-

Benefits:

In-depth subject specific knowledge, meaning that:

You will get started faster.

Your early mistakes and false leads identified and corrected earlier.

You will get a faster entree into the best collaborative networks.

You will get noticed by the community faster.

You will be associated with experts in the field.

Disadvantages:

Depending on the supervisor, you might get rail-roaded into doing the science his/her "way", leaving less opportunity for you to strike out on your own path.

You are dealing with a new advisor, and that relationship takes time to cultivate.

For your situation, it seems that you will have to get a source of funding.