Thank you for your answer, Christoph.

I finally manged to get it working. Here is what I did:

1. Created one CXF Endpoint in Control Panel > Configuration > System Settings > Web-API > CXF Endpoints
2. Created a matching JAX-WS API in Control Panel > Configuration > System Settings > Web-API > JAX-WS API, referencing my CXF Endpoint
3. Used the classloader from Liferay for creating my service:

Bundle bundle = FrameworkUtil.getBundle(this.getClass()).getBundleContext().getBundle(0);
ClassLoader classLoader = bundle.adapt(BundleWiring.class).getClassLoader();
Thread thread = Thread.currentThread();
ClassLoader contextClassLoader = thread.getContextClassLoader();
thread.setContextClassLoader(classLoader);

try {
    // create service
    return service;
} finally {
    thread.setContextClassLoader(contextClassLoader);
}

The first line (retrieving the bungle) was actually the crucial one. At first I tried to get it only via

FrameworkUtil.getBundle(this.getClass())

This returned "com.my.bundle_1.0.0" which is my own bundle, which is the service builder generated project. But actually I had to get the Liferay bundle via:

FrameworkUtil.getBundle(this.getClass()).getBundleContext().getBundle(0);

Additionally I should mention that my call isn't in an OSGi component, therefore I had to use this methode. If youre are actually in an OSGi component (@Component) then you can simply get the BundleContext in the activation method:

@Activate 
@Modified 
protected void activate(BundleContext bundleContext, Map<String, Object> properties) { this.bundleContext = bundleContext; }

JODD:

I worked around it by setting the target version in my pom.xml (Maven) to 1.8. Not a perfect solution but since I am not using any newer features I can live with it for now (and probably have to).