Well it looks like you sussed the problem. Exactly what happened with my old bitch a border collie, who was 13 when she started to get 'picked on', my first dog who got me into behaviour and training, so it was hard to see an up jumped younger collie stealing her position, but it had to be done.

The more I told him off for being mean to her, the worse the problem became. I should have just left them to it!
If you are sure you have got to the stage where you could safely 'up' Pepper, then it is quite a simple process, trying to use the natural behaviours of the dogs anyway, reinforcing them slightly in the favour of the promoted dog in you can.
At any time there is a squabble, don't tell Pepper off if she goes at Max, but just stop the squabble by distracting, then don't make a fuss of Max when you feel sorry that he had been picked on - this is the hardest thing to do in my book - Anytime there is an option of one being first i.e. doorways, getting in out the car etc. let Pepper take the lead, she may be trying to anyway and you haven't let her, if you have both sat and are feeding treats, reward Pepper first etc etc.
If Max has any privilledges that Pepper doesn't - one of our dogs is allowed upstairs at story time to sit on the bed with our daughter for a bit - otherwise they aren't allowed upstairs, then try and give Pepper a bit more time too. If you think through your routine there are probably numerous occassions where Pepper is already pushing in, so let her.
If you decide to go with it and help the situation along a bit, do be careful, as if you think Max won't back down gracefully, you could get a bit of scrap. Ideally let them do it themselves, with a bit of subtle engineering from you and it will all come good in the end.
Best of luck, it is so sad to see an older dog getting old, but it is what happens in every pack eventually, as long as Pepper doesn't upsurp you!
Kate