I've never separated a bonded pair when one has been neutered. And never had any problems as a result. On the contrary, provided they had a good bond before the neuter, returning home to a loving partner is a comfort to them when they feel ill.
It might help to ask your vet to let you bring the partner into the clinic and maybe keep him there for an hour, in a hospital cage if you can, preferably with his partner, but if not, then nearby. That will transfer some of the clinic smells onto his fur. Bring them home together, preferably in the same carrier so that her anaesthetic smell will be transferred onto his fur, and then let them out from the carrier into your home together, making it a shared experience, coming home from the vet.
My first female Rosie was a terror for taking out her stitches, even though the vet used the technique where they are buried under the skin, she could still somehow manage to get through the covering medical glue and open up the wound, so there was no choice, she had to wear a cone. I cut it down so she could eat, but she couldn't clean herself and her partner saw to all of that for her, he groomed her as thoroughly as she would have groomed herself. I used to transfer the caecals for her, picking them up as she produced them and offering them to her by hand. She was a bit surprised it seemed at first, but took them readily.
Some years later after she died and her partner Mr H d'Bun was rebonded with very young Cinnamon, the pair of them stayed together after Cinnamon's spay with no problems. Cinnamon's next partner, Nutmeg was castrated when I adopted him so this wasn't an issue, but after Cinnamon's death Betty came along. She wasn't separated when spayed. After Nutmeg's death came already neutered Alfie so again not an issue with separation but after Betty's death came very young Fearne. She became sexually mature at just over 13 weeks, their bond was very, very new and it was in huge danger of breaking up due to her behaviour. Alfie was not interested in the games she wanted to play! She was spayed at just over 14 weeks as my very, very rabbit savvy was happy she was healthy, well grown and of a good weight. She sailed through the op, came home and lay down by Alfie's side. That was that, no problems at all, and 3 years later they are still very happy together.
Potted history there going back over 20 years, but I have yet to experience a problem keeping pairs together after a neuter