The entire point of getting the Otter scroll for Asuma is so that he can get a political win. Akane getting it would not help his political situation, especially since it won't make us any less likely to commit treason (nor would it make us much more in his corner, since we'd be manipulating him into doing it).
Instead, we should be figuring out how to make sure it doesn't go to a conservative clan. Ideally, we want it to go to a moderate clan or Sand/Mist. The scroll going to a moderate clan will help Asuma the most politically without it going to the Hyuga/Hagoromo, and would give us an opportunity to get on better terms with that clan through the summoner trade network. It going to an allied village lets us get instantaneous communication and trade with that village, cementing our alliance for decades to come and opening the door to more trade deals between the Gōketsu and foreign clans.
The scroll going to a conservative clan would naturally be a disaster for us because it would interfere with our background efforts to turn Neji into a moderate or even someone who is sympathetic to our block. It going to one of our allies isn't the end of the world, but it doesn't give us as much benefit as the scroll going to a moderate clan or an allied village.