Managing a player like Reina if (as it appears) we are having trouble moving him on is part of being a good manager. I don't buy that he wouldn't return to the fold if he and BR had a proper chat and put their cards on the table.
It would come down to fairness, if BR tells him straight-up that he will get a chance if Mignolet screws up and that both will be treated equally I see no reason why he wouldn't accept it and get on with it.
He was easily as good as Mignolet last season at Napoli and his skills are far more suited to the way we play than the current Mignolet. If the pair of them push each other to another level it will be great for the club.
Get the lad back, bring him into the fold as with Suarez and get him back to his best. Rogers is a master at it with every other player so why not Reina.
Totally sums up my opinion. Let them duke it out for that spot. Reina's shot stopping needs work, and Mignolet's command of the box/distribution needs work. I think Mignolet has more raw talent and potential, but Reina has more experience, more football IQ, and more tools (currently).
Rodgers and Reina are both big boys, and I don't think this media tiff would really be that much of an obstacle.