I think it would work much better as a middle name, if you really insist on using it.
The problem with River is that it lacks foresight. Your daughter won't be a little girl forever. Do you really think that River is going to age well? When she's 40? When she's 60? When she's 80? There are plenty of names that might sound cool when the person is a child, but it's important to pick a name that will sound good for the rest of their life - and that will suit her no matter who she grows up to be. River is very specific, it really wouldn't suit most women. You can't predict what she'll be like, so it's important to pick a name that she can work with.
That being said, I think it's okay to go a little unusual with middle names, since the person won't be forced to use it if they don't want to.
I actually think that Athena River would flow better (no pun intended) than River Athena. I also think that Athena is a lot classier and that it's more likely to work when she's a grown woman. I can picture a little girl, a grown woman and an old lady all plausibly being named Athena. Plus, it stands out in a good way. There probably won't be anyone else in her class named Athena.
I definitely wouldn't do River Rose. Or anything Rose, considering the fact that almost every little girl has Rose as her middle name, these days. Aisha is a nice name, but I don't think it flows. I do think that Sienna River would work, too, but it's not as nice as Athena River.