As for competition between players, I don't think so because Real Madrid can always have quite solid players and if competition occurs then it could have happened long time ago when it was still in the Ronaldo era where Real Madrid also had many great players.
Moreover, currently Real Madrid has players who overall can play very well and have teamwork that cannot be doubted, just look, even though they are young players, there is still no competition between the players.
We don't need to worry about this, although I admit that Mbappe is player who is quite stubborn and always wants to stand out.
That should not have happened and Ancelotti is also an experienced coach so he can certainly make problems like this easier. Evidently what is done by Ancelotti is always successful and he has also been so experienced in handling Real Madrid that it will be much easier to determine players. Having a star player in the squad can also make it easy because he can rotate every time needed and will be much easier when some players are injured.
I don't think the problem will make Ancelotti dizzy because he already understands what to do. Now the big responsibility is how Ancelotti can defend the La Liga and Champions League titles next season. Mbappe's presence should be a force that will make it easier for him to defend the title.
I believe Ancelotti is quite firm and he prioritizes the team over individuals. Mbappe is known to have a bad attitude when he was at PSG, he was too powerful without thinking about the interests of the team, which made PSG slump in the Champions League.
At Real Madrid, Mbappe really has to get rid of that kind of attitude and Madrid is a team that has a long history of winning the most titles in Europe. Of course, it is hoped that Mbappe's arrival will help Madrid continue to defend its title and I am very sure that Mbappe's presence will have a tremendous impact if he remains professional and respects his teammates.