Go to your moodle settings:
Default is 8 hours in a 3.11.
Comment on that setting:
"If people logged in to this site are idle for a long time (without
loading pages) then they are automatically logged out (their session is
ended). This variable specifies how long this time should be."
But that's just half of the problem ...
"How high can I raise the SMTP session limit without any additional risk to being marked as spam/blacklisted, and will raising this limit help with this issue"
Your setup might have robust ability to send, but what of the mail servers receiving? You don't control their settings/checks for volume of message coming from your AWS setup.
For example, you have 300 users who get their email from somenet.org ... somenet.org thresholds for number of messages in one session is lower than gmail.com. And you can't fix that! :|
And somenet.org also submits your AWS relay to blackholes as a result. You don't really control that either!
Go to: /admin/search.php?query=smtp
At the bottom of that form:
SMTP session limit
"Maximum number of messages sent per SMTP session. Grouping messages may speed up the sending of emails. Values lower than 2 force creation of new SMTP session for each email."
Yes, it's a problem ... one for which I don't think there is any one single answer, unfortunately.
In Scheduled Task of your site there is one for:
Forum mailings and maintenance jobs
You might have to resort to setting that up to allow 'run now' (path to php is required) and use that.
There are also 'adhoc_task' ... which catch portions of the cron job that didn't complete for some reason and will be attempted again.