I don't even think he has plans to let him stay, well this is my own opinion and from what I have seen at least from the few games Alonso has been in charge. Rodrygo wasn't injured during the Club World Cup but for some reasons been known to the coach he didn't feature him. He played few minutes coming in as a substitute and I don't think he has started any of those games, this is a players who was a regular starter just few months ago. Personally, I don't think the coach wants him.
They didn't play many match on the world cup I think it was just three but Alonso isn't really that mean to him in the extent of not wanting to play him regular or at least starting ones on the matches, so actually Alonso did started with him on there first match against AI Hilal, Rodrygo was there number 11 on that day having Garcia and Vinicius with him on the forward but that was his last time of starting any matches but I understand why Alonso did that but he actually tried because even if Rodrygo might not be at the first but there isn't any matches he did not put him.
Real Madrid went as far as the Semi final in the club world cup and by contrast that way above 3 matches. Round 16, Quarter finals and semi final that's a total of 6 matches they played in the competition. You were right when you said he started the game against Al Hilal and that was his only start in the competition the two other games in the group stage against Pachuca and Salzburg he came in as a substitute which is not that bad but after that I don't think I saw him again. Well he might not fit into the coaches plans.