Some thoughts, independent but not necessarily mutually exclusive...
We know that there is a technology that runs off of, well, "blood magic". But does conventional technology exist as well? The Unblooded -- for lack of a term -- could do their own research and develop a parallel technology up to whatever level works for the story. Maybe crossbows, maybe guns. Could even be you work these discoveries into the story.
A different angle. Perhaps some researcher (from either group) discovers an antidote or suppressant for the blood chemicals. This is your world's equivalent of mustard gas. Drop the gas into the battlefield, then a special squad of Unblooded can swarm in with their low-tech weps and reap a bitter harvest of the panicked enemy whose magic stopped working. Eventually word will get out and enemy armies will have the same technology, and you'll have some very confused battlefields as the presence of gas and wind strength give shifting advantage to Blooded or Unblooded.
Third way. As others have mentioned, your hero could be an infiltrator or special-ops kind of guy. He could specialize in assassinations, poisoning water supplies, or all kinds of mischief.
Update: Forgot to address how this blood magic works. If it's just plain "magic", then we're done. That's all you have to say. But let's get just a little sciencey here.
Could be their blood has lots of alcohols in it, which can burn to power devices (this also explains why the guys without a high blood alcohol level are more observant ;D ). Could be the blood has a special protein which binds with radioactive elements, again to power devices. Unblooded have not this protein.
In either case, a very interesting effect comes in to play. You said that technology runs off of this chemical. That means that it gets ... expended. It's a finite resource. So, how long can you power an exoskeleton before you start feeling all faint and anemic? If it's a short time, this gives another opening for the Unblooded on the battlefield. They have way more endurance than the Blooded. Normal Blooded battles must be short, as both sides will run out of blood chemical. Imagine if you kept a small, angry reserve of Unblooded waiting for the main fighters to grind down before they pounce...