That works for unit testing a specific component, but not so well for testing your system end-to-end.

This! To me the only upside to the whole architecture (from a devs perspective) is that you can deploy these things to production independently of other parts of the system. If you can't do that with confidence because you're attempting to test it's functionality in some larger deployed context you've turned your monolith into a distributed monolith and now you have the worst of both worlds.

The good news; you should be able to accomplish most testing locally, in memory. The bad news; your test code is probably going to be much larger and your going to have to be very aware of the data model representing the interface to the lambda and you're going to have to test the different possible states of that data.