Running a task without waiting for the results in Postgres 11 on RDS
You seem to be beating your head against the limitations of RDS.  Is what RDS offers you worth the limitations, over just renting the hardware and running your own database on it?

Triggers and NOTIFY/LISTEN are for event-driven things, not clock-driven things.  They are for solving a different problem than the one you have.

My prefered approach is to combine 'cron' and 'Your language here'.  Write the job in your language, launch it with cron.  For example if I am writing a web-app, I actually write the maintenance logic into the web-app, and then have cron use wget to hit a special URI to invoke it.  Yes, that would require having a non-RDS machine that lets you run cron and your language of choice on it (unless you give up on RDS), but if you have a web-app, you already have an app-server which could be used for that (unless that machine is also AWS-managed with no CLI access).

You could also use pgAgent in the same way.  You would need a separate machine to run the pgAgent software.  You could also run a tiny database on that machine to hold the jobs--pgAgent can read the jobs from one database but execute them against another one.  I think (but I have not tested this) that you can even put the jobs into a schema of your main database.  While you cannot use 'CREATE EXTENSION pgagent', you can run the script manually to create all of the objects.  (And if you do want to use pgAgent, you could bug AWS to whitelist it--since it does not require a superuser there is no reason not to do so other than the effort involved)