Kevin, I think your group is somewhat common these days, based on what I've seen as I've done programs about the RAF around Florida. Mostly older folks, aviation enthusiasts of one stripe or another, but lacking a commitment - for whatever mix of reasons - to be active pilots. One of our local EAA chapters mirrors your club: We routinely get 20+ members at a monthly meeting but there are only 5 or 6 active pilots...and I'm being generous when I use the word 'active'.
The critical issue seems to be your chicken/egg dilemma of wanting to grow the club via an LSA purchase vs. the cost of the purchase - and its relevance - for a small, mostly non-flying membership. Your original notion - buy an LSA and they will come - may be sound (or just overly optimistic), but it's hard to imagine how the numbers will work. I'd suggest you initially focus on the club and not the aircraft. Your options seem to be to grow the club, thereby expanding its options, or let it remain as it is and simply remain a social member of a mostly inactive club until the membership dwindles away on its own. There are many ways to energize the existing group while attracting new members, and your challenge is to try some that would be suitable for your group and the local aviation scene. Some might be programmatic in nature: group tower visit, flying contest, hosting a ground school for local, interested H.S. students, guest speakers, importing AOPA's 'rusty pilot' program, picking one improvement project for the 172 and tackling it as a group - the list is almost endless. Others might be more civic minded, perhaps an improvement project at the airport, or hosting an airport open house, or raising funds to sponsor a deserving student pilot.
I mentioned I've seen many such groups as your club...but I've also seem some real success stories. One excellent example (and very much aligned with your current situation) is Friends of Arcadia Airport -
http://foaa.us/ This group has rescued a municipal airport essentially abandoned by its city govt., they have now created an excellent 'Fly in/Camp out' oak-shaded camping area for visiting pilots, they host hugely successful Saturday breakfast fly-ins and an annual 3-day event tied to their local professional rodeo, and they are building Florida first pilot shelter with a large pool of donations they've raised. Yet only a few (3? 4?) of the members are active pilots. The leadership chose stepping stone projects that local aviation enthusiasts could get behind and they made sure those projects had a social component, as well. I've seen other examples of this same circumstance - e.g. a small group of leaders rescuing an increasingly inactive but historically famous Florida club who's assets were being lusted over by a few others. It is now flourishing.
I think your first challenge is with your club and its identity & purpose. That strikes me as the dog. The aircraft is the tail.
