The best way I've found to do this so far is using the Mixmax addon. It'd be ideal not to have to install an addon, particularly because of the extra bloat it adds, but it seems easy enough with this addon to ignore everything else ;)
It specifically supports Inbox (as well as vanilla Gmail), and has the exact signature option I was looking for:
It seems to be quite nicely designed and coded as well, which is a plus.
One major downside: it adds this footer to the bottom of every e-mail you send:
It seems you can remove it on a per-email basis by clicking the x, but removing it automatically for every e-mail requires upgrading to a paid plan.
The footer doesn't bother me much particularly because it's underneath my signature and wouldn't be noticed by many, but I understand why it would bother some.
I'll update this answer if I come across any better ways of placing the signature above the quoted text in Inbox.