Do I really need an H-Bridge controller in this application?
One way or another you need to reverse the polarity of the pulse applied to the valve coil to change its state.  An H bridge is an easy solution, especially because there are self-contained H bridge driver chips that would be capable of driving this valve without needing external FETs.

There are lots of other ways you could do it.  For example, if you had a split (+ &amp; -) power supply then you could connect one end of the coil to ground and switch the other end to either the + or - supply, but by the time you're done with the level translation and so on this seems like far more work.  I'd just go with the H bridge unless I was really in it for the creative circuit hacking and not simply the least effort.