the club would have had to release one of their existing foreign players

[Maybo]

In order to include Ronaldo in the squad, the club would have had to release one of their existing foreign players and they were aware of it while signing the Portuguese talisman, the report stated. (Aboubakar leaves Al Nassr on a spicy note by Sports Desk)

I have a problem with the tenses. Why “would have had” was used but not “would have”?

 

[Tarheel]

Say: "Why is it 'would have had' and not 'would have'?"

In fact, since they did, in fact, sign Rinaldo, it should be "had" and not "would have had" there.

When you are trying to meet a deadline you are more likely to make mistakes.

 

[Maybo]

But I found that they signed Ronald but they haven’t registered him. In that case, is “would have had” correct?

 

[Barque]

But I found that they signed Ronald but they haven’t registered him.

It's nothing to do with registration with the league. If they have signed Ronaldo (executed a contract with him), they will definitely register him (with the league, as one of their foreign players), or they won't be able to make use of him.

Why was “would have had” was used and but not “would have”?

"Would have had" isn't wrong there. But it'd probably have been more appropriate to a situation where they'd decided not to sign Ronaldo up.
They didn't sign Ronaldo because they would have had to release an existing foreign player to do so, and they didn't want to do that.

But it also works in that sentence because it refers to the situation at that time.

In order to include Ronaldo in the squad, the club would have had to release one of their existing foreign players and they were aware of it while signing the Portuguese talisman,

I'd have worded it differently: The club knew they would have to release one of their existing foreign players when they signed the Portuguese talisman.