Historically, we have dealt with belligerent non-sovereigns (e.g. pirates) via the military. How do you think treating Al Qaeda as a military threat has corrupted the military? My great worry is that if terrorism is treated as a crime, then the protections available to all accused criminals will be watered down to accommodate the needs of prosecuting terrorists. That's what, e.g., has happened in the drug war. There are a lot of bad doctrines (e.g. no-knock warrants) that were motivated by the drug war (specifically the ease of concealing or destroying evidence of drugs).
By blurring the line between law enforcement and military actions we've made it much to easy to deploy military force (most notably airstrikes) as a means to expediently solve problems. Where does that slippery slope end? When do we start carpet bombing drug cartels? We've already been invited to do that.
War doesn't have to between sovereign states. War against non-state actors is accepted as valid.
The US constitution even has provisions for it: Congress can call forth the militia to suppress insurrection.
The Taliban was a sovereign state that, for a basket of reasons, actively supported terrorist attacks on the US. The invasion of Afghanistan was a war.
Al Qaeda is not a sovereign state; they're a criminal organization.
What makes it difficult to prosecute Al Qaeda using US criminal procedure is the fact that they're mostly non-citizens, mostly operating out of the US, mostly in countries that are not especially cooperative with the FBI.
In fact, I think my counterargument says that the "grave problem" is the opposite of the one you pose. It's not that prosecuting Al Qaeda criminally risks corrupting the criminal justice system. It's that prosecuting them as a sovereign who we're at war with has demonstrably corrupted the military.