I have done work integrating with JIRA. Here are some of my thoughts…
I have found no way to do what you are looking for given the current integrations provided by monday. Since the item ID doesn’t change when you move it to another board, it seems like it would be relatively easy for monday to remove that limitation. I think it would also be possible using JIRA webhooks to create your own “integration” with something like Integromat. It would be quite a bit of work.
The direction we are moving is to have one board that is the JIRA representation in monday. We are then using filters, views and groups to give us the items / look we want. We have one situation where an automation is setup in the “JIRA” board to create an item in a different board then use mirror columns to give the JIRA visibility in the second board.
Also, though not directly related to you need, we have put in place processing using JIRA webhooks and Integromat to deal with the situation where JIRA tickets get moved to a different project.