Barcelona will never be able to compete with Real Madrid in their current state without Mbappe. There are many differences between the performances of these two teams, Real Madrid and Barcelona. Real Madrid is the best performer in every tournament as well as they are performing great in their league and their performance is consistent in contrast to Barcelona but they are not performing well in all tournaments and Barcelona performance is not consistent.
Real Madrid are signing Mbappe to further strengthen their squad so that they can beat the big teams in major tournaments. The main reason for Real Madrid signing Mbappe is not to compete with Barcelona or insult Barcelona, but to improve their overall performance.
Barcelona has not been a problem for Madrid this season. Even with the current squad I think Madrid is still capable of winning the La Liga title next season. So I think you are right that Madrid's goal in recruiting Mbappe is to improve the team's overall performance. With Mbappe, Real Madrid will be the best club in the world next season. And I'm sure they will be targeting a treble winner or even more next season. With Barcelona's current financial condition I think they will really not be able to compete with Madrid next season.
Not just Barcelona, no team can compete with them anymore. Real Madrid will rise to a different level after Mbappe joins the team. Even the rotation players will have great talents in this great squad of Real Madrid. It seems very difficult for Barcelona to compete with this team given its current economic conditions.
Don't compare a club that has so much money with Barcelona which is experiencing financial problems this season. Barcelona can't afford top players, but they are aware that last season they were able to win the domestic league even though they didn't have top players or really well-known young players.
In fact, I am very worried about the arrival of Mbappe in the Madrid squad, because Real Madrid once had top players in 2003/2004 where Real Madrid was filled with star players, however, they had difficulty being called a strong team and they were even eliminated by Even though some clubs have classy players, if there is no unity in a team it will be a waste of time.