While I definitely think it's a travesty for Beck to end up lost in the outer rings and would never consider going to an arts and culture event at a lifestyle center, I think the Beck Center finds itself in a very precarious situation. Given the current financial conditions facing our county's arts and culture nonprofits (together with the abundance of organizations currently engaged in or launching capital campaigns), it would certainly be enticing to move into a brand new facility that could better serve your patrons. Moreover, it is my understanding that the vast majority of Beck patrons reside in Bay, Avon and Westlake, not in Cleveland or Lakewood. And unfortunately, Beck's mission ("To be a comprehensive community arts center focusing on theater, arts education, and gallery exhibitions") doesn't tie it to a particular community.
The obvious losers are the people of Lakewood and Cleveland, who will have considerably less access to the educational and outreach offerings of this institution. But the outer ring suburbanites, in my opinion, lose out as well. Arts and culture bring together a more diverse group of people than any other human social activity. If we start dividing up audiences by income bracket, level of educational attainment or even address, I think something about the experience gets lost.
So, just like any other business or nonprofit vacuumed out of Cleveland by urban sprawl, if you care about urban arts and culture organizations staying urban, then it's important to support that organization in its current location and to get other people to do so. When we don't, it's just too easy for a cash-strapped organization, even a charitable one, to follow the money toward (or even beyond) the county lines.