Ask Lambden Milligan what judicial review there was.
Heck, the US courts couldn't/didn't rein in Lincoln or his military (until after Lincoln was dead). I'd be careful about being smug, if I were you.
But judicial review was available and defendants did take advantage of it. Avenues not available to southern political prisoners. You knew that and you call me smug?
Can Civilians be Tried by Military Courts?
As Neely pointed out representatives of the confederate government like Thomas Hindman ruled by fiat and ignored local courts at will. And the habeas corpus commissioners ordered people jailed without trial, military or otherwise. Since trials, military or otherwise, didn't occur then the question of whether civilians could be tried by the military courts was somewhat moot.
Like the review that was available to John Merryman in the North in 1861?
The US Congress did not authorize Lincoln to do so until 1863. Why would the US Congress need to do that if Lincoln had the right to suspend habeas corpus in the first place.
The Confederate Congress authorized the suspension of habeas corpus on several occasions. When habeas corpus is suspended by the government entity authorized to do it, you could appeal to the courts until you were blue in the face. The courts would rule that habeas corpus was legally suspended, so why bother?
Thomas Hindman was removed by Jefferson Davis for his restrictions on civil liberties.