The only way we were able to get this to work was to set each unique email address up as a Distribution list with the Lansweeper mailbox being checked via the graph api as a member. Lansweeper then respected the to: field action in the ticket dispatching rules. It simply would not process the emails to: field correctly if it was an alias to the Lansweeper mailbox - even though the headers preserve to: as the correct aliased email.
This unfortunately created a new issue however as now the ticket-dispatching rule email address (aka the distribution group) is now added to every ticket as a CC user, which will eventually cause issues.
I did open a ticket shortly after posting this - the response was lackluster. They acknowledged the problem with the CC user, completely ignored the original problem (sweeper not honoring the to: field), and offered up only to put in a feature request. Considering Sweeper has stated they aren't developing the on-prem Helpdesk product any further I'm not really sure why that would have been the solution/resolution to an ancillary problem while completely ignoring the actual issue.