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.
Real Madrid had only 1 defeat this year, they are performing really well in the league. If they make one or two strong transfers next season, they will be able to really fight to be the best in the world. I think Real Madrid's current performance in the league is very good, and they can become undefeated champions after strong transfers.
In this case, Barcelona will have to work to finish the season second again

Ancellotti and perez are working now on the big project of real madrid.
Spanish outlet Defensa Central are reporting that Real Madrid president Florentino Perez has promised the signings of Leny Yoro and Alphonso Davies.
Source:
https://www.football365.com/news/real-madrid-kylian-mbappe-confirms-transfer-announcement-plan-perez-promises-two-signingsEven perez has promised two signings to come first. Signing alphonso davies has indicated that if real madrid is now seriously rebuilding its defensive line again. Real madrid needs to recruit good RB since the current RB is not that good. Someone like TAA will be very beneficial for the club.
Madrid has targeted recee james to be recruited by the club but he rejected it. Things are moving forward, and Madrid is now interested in recruiting TAA. Im expecting another plan if TAA will also refuse real madrid's offer.
Madrid has very good attackers, these attackers can be considered as the best in the world too. Madrid is only lack of defensive line which is also becoming the main problem. It has very less GF compared to the barcelona but the defensive line can sometimes still make blunder.
The defensive line shall become the main concern by real madrid to build the perfect team again. Im hoping that madrid will able to get a good RB for the club.
I can imagine how strong real madrid if it will have some one liek TAA or reece james. Im still waiting for another player probably will be signed by real madrid.