Has anyone figured out how to deal with "dovecot" email handling on Siteground?
I presume that you are talking about bulk mailings. I have got this working for small volumes hosted on Siteground. I had a lot of problems originally with subfolders though can't remember the details. My solution was as follows.

Outbound use mail() and in Administer>>CiviMail>>Mail Setting, set Mailer Batch Limit to 100 and Mailer Cron Job Limit to 1. With the Send Scheduled Mailings cron job running every half hour this throttles the outbound mail within SiteGround's limits.

Inbound i set up an account on GMail to handle the bounces so bounce processing account (Adminster>>CiviMail>>Mail Accounts) was set up using IMAP to GMail and address examplereturn@gmail.com and localpart return+ and email domain of return.example.org. I then forwarded @return.example.org to examplereturn@gmail.com. The GMail account needs some special settings as described in https://docs.civicrm.org/sysadmin/en/latest/setup/civimail/inbound/#return-channel-google-apps. All rather complicated but nothing else I tried worked. 

Hope that helps.