'Would have shot' is in the conditional perfect tense, which is the correct tense in this case.
The base sentence, showing this tense in the third conditional would be:
If you were my enemy, you would have shot me.
Negating this sentence gives us:
If you are not my enemy, you would not have shot me.
A slightly more complex version of this sentence could be:
If you are not my enemy, I don't think you would have shot me.
In your case the writer has split both parts of the conditional sentence into two separate sentences. This is not usual, but it is allowable. In effect, this now becomes:
I don't think you are my enemy. As a consequence, I don't think you would have shot me.
Add in a few more words for dramatic effect, and you end up with:
Not that I’m saying you’re necessarily an enemy. I mean, I don’t think
you would really have shot me.
The use of 'would' in the second sentence shows that this sentence is in the conditional mood. The use of 'have' shows that it is in the perfect (ie past) tense. Removing 'have' would change the sentence structure from conditional perfect to simple conditional, which puts it in the present tense. This would require the verb 'shot' to be replaced by the verb 'shoot'. So your amended sentence would change from:
I don't think you would really shot me.
I don't think you would really shoot me.