JIRA Rest API authentication w/o 3lo handshake

We develop a worktime-measuring system that has projects and tasks similar to the Jira projects and issues. I made a synchronizer tool for synchronizing the Jira issues to our system. After the issues synchronization the users register their time on our system then the tool synchronizes back the worklogs into the Jira. This tool works perfectly on Jira Cloud, uses the Jira Rest API through JWT authentication with impersonation (for the worklog registration). We created an atlassian-connect add-on for this. I found that the atlassian connect addons doesn’t work on a self-hosted Jira Server. I would like to find a way similar to this scenario that works on self-hosted Jira, but I can’t find a way to implement the authentication. The 3LO OAuth authentication doesn’t work for us, because we can’t ask every single user to interact with the browser and collect the auth data. This tool could run on the same machine as the Jira Server. I’m interested in every kind of solution.