If we don't get him to the hospital soon, he may bled to death. (He may die shortly)
If we didn't get him to the hospital quickly, he would have bled to death. (he is alive now)
If we didn't get him to the hospital as soon as possible, he would have bleed to death. (he is alive now).
Sentence 1 no Have is required.
If we didn't get him to the hospital as soon as possible he may HAVE bled to death.
Didn't in the past has HAVE in the sentence.
Don't in the present tense doesn't have had in the sentence.
Booth good but sentence 2 is describing it as if it was a live situation now and ongoing. If you are at the scene and you have to take/get them to hospital quickly, then sentence two is good. You would say this to somebody else at the time.
But to talk about it after you have to change it from soon to something different as per the sentences I showed you.