How to combine BPM and Scrum?

The idea seems to make sense, as you mention, there are different roles and tasks. But the management of a project is, in my opinion, more like a Case (see Case Management Modeling Notation (CMMN)) and not a Business Process in the standard definition of BPMN fixed activities, input, output, and gateways according to those outputs.

Also, if you follow a relatively flat hierarchy, you will not have so many roles in your team and the work will be more a collaborative effort than a "single assignee" per task.

This being said, there could be situations where a more structured BPM process can be applied, but typically don't, as far as I have seen. I can imagine a scenario of using Scrum and BPM maybe in other industry different than software engineering.

There are some Scrum practices that BPMN could easily enable for example daily meetings, as a BPM process can give us the tasks in execution, what was executed yesterday, what can be probably executed tomorrow, and by whom. But again, this is also solved by Project management tools with Kanban support In any case, these kinds of processes should allow the inclusion of ad-hoc tasks on any step of the flow.

So, even if is possible, if we are talking about a pure BPM process, the management of its execution requires an approach more closely to this discipline than Scrum, because is not focused only on the execution of particular tasks but in the improvement of the process itself, through design, execution, monitoring and improving, what is called the BPM lifecycle.