She is pretty particular in the way the horses are introduced. First off, we almost never have mare and geldings turned out together. It is just more trouble than it is worth to deal with that.....although many of us (myself included) have kept mares and geldings together succesfully, they are trickier to pair up. Typically the mares are more high strung and dominant, exactly like what you are describing.
Second, the horses are introduced slowly. The herd members have their stalls together in the barn, so the horse going into a herd will get a stall next to their new potential herd mates so they can visit. During the day, she will use the outdoor arena, which has a pipe corral attached to it, to let the horses continue to socialize. The new horse goes in the pipe corral and the others hang out in the arena. She watches them and if they seem good, the next day they all go into the arena together. That way they can be watched. If they start really getting into it, help is close by, and she and some of the staff will run in and separate them as soon as it is safe, and they will try the new horse with a different group. If they get along reasonably well, the following day they go out to their regular turnout. BTW, we have a huge covered arena and miles of trail access, so people don't mind when the outdoor arena is used to introduce new horses.
If this sounds like a lot of trouble, it is because it is , but it is the safest way to manage new herds and changing herds. My BO will not hesitate to change the herd groupings if a horse is getting picked on, and she can tell you what positon each horse holds in their particular herd because she watches them so closely. She is a gem.
The bottom line IMO is that horses can get badly hurt or even killed in the wrong herd, and it is too risky to keep horses that are not getting along, together. I think your horse needs to be safe, and that means probably moving the mare.