当应用程序安装在真实设备上时，“此应用程序将无法运行，除非您更新Google Play服务”(“This app won't run unless you update Google Play Services” when app is installed on real device)

This android developer's blog entry describes what's new in the version 7.0 of Google Play services.

GoogleApiClient

In one of my projects i use this

compile 'com.google.android.gms:play-services:6.5.87'

It's the whole package and I'll pick only the modules I need before deployment. Works fine against target API 22.

The farthest version I'd go with would be

compile 'com.google.android.gms:play-services:6.1.71'

GoogleApiClient

So should I use an older SDK?

Absolutely not, this is not an SDK related problem. Keep everything (build tools, compile SDK, target SDK) to 22.

is there a better way to handle this than asking my users to update their Google Play Services?

See above, use older version of the library with your project.