Answering my own question:

...After some thought it became apparent that I was right. ...I did have all the pieces.

z = a*sinh(u)*sin(v) (As per the wikipedia page)

At v = 90 (the z axis) solve for a

z/sinh(u) = a

So given a proportion (which is just a modification of u -- for my purposes i just create an arbitrary u anyway) I have my "a" variable.

I hope this post can be helpful for anyone searching for a similar solution at some point. (It's already in the top 2 results for "Random point in a spheroid").

Please note--even distribution is beyond the scope of this post. ...Gotta figure that one out for yourself.