I guess I just don't see that attitude in too many places. We're going from running a small food pantry (inefficiently) to hosting Northwest Indiana Food Bank, a secular charity that needs a building with a parking lot like ours and some able bodies to load food into cars periodically, along with whatever donations people offer. They aren't Lutheran or even religious, nor are they a government program. She'll use whatever church has a suitable facility, and they do all the work of screening the clients, working with wholesalers to get perishables, and letting the clients know when and where to ick up the food. All we have to do is put out cones for the line of cars, direct traffic, load trunks, and do so without blocking any streets. So if the goal is to get food to people who need food, you can't do it more efficiently than that. But I guess we're cooperating in externals with atheists and pagans and anyone else who might donate food to a food pantry.
Many conservative LCMS churches work with 40 Days for Life, founded by an Evangelical with events featuring mostly Catholics and Evangelicals. Or Crisis Pregnancy Centers, same deal. We host AA, Al-anon, and Teen Al-anon in our building without any religious tests. The only time people bristle about working together is when something presents itself as a Christian thing and the togetherness aspect is the real point rather than the food distribution of help for young mothers or whatever. When people who don't think our differences are church-dividing invite us to an "isn't this neat that we're so united despite our differences" event, we generally demur.