Buddy sour is not a fun problem to deal with. Hopefully the people at your barn are understanding because it can be a noisy problem to fix.
A lot of how you want to solve the problem depends on your individual situation. For instance, you can "wean" the horses from each other completely, but you might rather keep them pastured together.
It sounds like they are together pretty much 24/7, and even when you ride the pony it sounds like the two of them can still see each other.
I like my mares to be pastured together and I like them stalled near each other for my convenience. However, this means they get clingy from time to time.
For me, I make sure I separate them regularly. I take one into the barn to work with and ride while leaving the other out in the pasture. At first when I did this they screamed for each other. The one outside would run up and down the fence and the one I was working with would prance around some. I completely ignored the behavior and did not make any allowances for it. I rode them and did whatever I wanted as if they were behaving normally. Eventually they accepted the fact that they have to be separated sometimes. Now they are calm about it. The other boarders were nice about listening to all the racket while my horses were getting used to being apart.
It can help to move one of them down the barn so they are not stalled near each other. At first if you can't ride one away from the other, just lead one far away and make them stay apart for awhile until the horses get used to it. Bring a different horse along to ride with so your horse gets used to others. If you ride the two that are buddy sour together, practice having one stand while the other rides a distance away and then comes back.
Whatever you do, don't give into the temptation to not separate the horses because it is easier. You will be sorry when a situation comes up where they have to be apart.