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.
We can definitely consider the current Real Madrid without Mbappe as one of the best clubs in the world because of all the clubs in Europe and outside of Europe, Real Madrid will definitely be in the best position in terms of performance and winning trophies. Last season Real Madrid had players like Karim Benzema and last season Real Madrid were definitely favorites to win the Champions League title but they lost to Manchester City in the semi-finals. Real Madrid have a good chance of beating Manchester City this season as they lost to Manchester City last season and will face Manchester City again this season in the quarter finals. Real Madrid is already a strong team, so if Mbappe joins Real Madrid again, Real Madrid's strength will increase.