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.