First off, root does not need to be in the dialout group since traditional permissions allow root access to the device. Non-root users need to be in this group (or ACLs used) to access to the device. The snap is not in a position to grant this access either, except with the account-control interface and then only on Ubuntu Core. Also, strict mode snaps intentionally do not have the access for using sudo.
Today, the only way is for an admin user to grant the access manually, eg: sudo adduser <user> dialout, but again, the snap cannot do this. It is technically possible for snapd to provide a mechanism for snaps to request permission changes on specific files, but that needs design and isn’t currently on the roadmap.
There is a specification that deals with some of this here: use case 2. When that is implemented, it won’t give your snap the ability to add users or modify groups, but does allow for later work to build upon it, perhaps something that would allow a gadget to assign users to groups.
As for octoprint, I reviewed its snap.yaml and see nothing in its security policy that would allow the snap to change group memberships. Perhaps it detects the error and gives the user helpful feedback on the command to run to allow access for non-root users (this is what I suggest to you as well).