Here is a link to the NCAA Division 1 Manual. You can download the pdf version for free.
http://www.ncaapublications.com/p-4284-2...aug-1.aspx
I believe the relevant rules are as follows:
20.02.5 Multisport Conference. A Division I multisport conference shall satisfy the requirements of this section.
20.02.5.1 Minimum Number of Members. A multisport conference shall be composed of at least seven active Division I members. The member conference shall include at least seven active Division I members that sponsor both men’s and women’s basketball.
I believe that this language implies that a school cannot be a full member of a conference unless it plays men's and women's basketball in the conference since a conference could not count such a school as a member for purposes of meeting its membership requirement. This does not mean that a particular school can't be an independent that plays basketball in a conference as an associate member or that plays any other sport in a conference as an associate member. It is also possible that by playing men's and women's basketball in the C7, a school like Gonzaga could be a member of the C7, but could play many or all of its other sports as an associate member in another conference.
Keep in mind that many sports have no conference regular season, such as track, cross country, tennis, and golf. Gonzaga could easily participate in C7 championship events in this sport as they require a single road trip (with no required return road trip by other C7 members). Only team sports require multiple trips. Looking at Gonzaga's website, this would implicate only baseball, men's and women's soccer, and women's volleyball. Gonzaga may be able to place these in the WAC, for instance, on an associate member basis.
Having thought about it more in researching this post, Gonzaga suddenly seems like a more viable candidate.