Use with Moodle

I am trying to get eLTI to work with Moodle.

I have followed the steps in the "Implementation" section of the documentation and managed to deploy eLTI, without changes in code, to my AWS account. Setup and integration tests succeed.

Now I am trying to do the "Out-of-band registration" from the "Architecture" section. The URL to pass to Moodle for LTI Advantage registration, I believe, is what I see in API Gateway / LTIConfig / Stages, something like:
https://.execute-api.eu-central-1.amazonaws.com/prod/tool

This fails with {"message":"Missing Authentication Token"}. I have no idea how to address this.

I also tried manual registration, with the following settings:
Tool URL https://.execute-api.eu-central-1.amazonaws.com/prod/tool
Public keyset https://.execute-api.eu-central-1.amazonaws.com/prod/jwks.json
Initiate login URL https://.execute-api.eu-central-1.amazonaws.com/prod/login
Redirection URI(s) https://.execute-api.eu-central-1.amazonaws.com/prod/authorizerProxy

with the latter ones only being wild guesses, admittedly. This allows me to create an activity on Moodle, and when launching it I get {"error":"No PlatformConfig record found for PLATFORM#K3xsVRt2SFeFIJP#http://my.moodle.server#1."}

I see that this results from a DynamoDB lookup from a table that only has entries for lms-integ-test.com. I suspect some entry for Moodle should be added to this table, but probably not manually.

I would of course prefer the automatic registration to work.
Additionally, I believe it would be helpful if an example Tool using eLTI would be available, as I have little idea how that should be implemented. Any guidance or references to documentation would be appreciated.

Also, is it possible to request assistance as part of the support included with my AWS account, and if yes, what service should I indicate on the support request form?

Thank you very much!