I am really wondering why is has to be that complicated. Jira is there to help us getting things done and not to restrain us with unnecessarily complicated workflows and fiddling with labels or personal messages. Moodle has such a nice development process, it only lacks straightforwardness for people who don't have full power in Jira.
In my humble opinion, I don't see any reason why there has to be a permission separation between the "Start peer review" and "Request integration review" transition in Jira. Anyone who has proven reliable enough to develop for Moodle should be able to push patches trough to integration review without the help of a superpower user. It would be easier to retract development permissions from anyone who falls from grace and abuses this "Request integration review" transition completely than to delay valid peer reviews of motivated peer reviewers.
Just my two cents,