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.
There is no doubt we have too many problems for the Barcelona on and off the field, but still recent results are showing they are recovering from their bad period and things are going on track even still it's very long way to go for them because debit and few other issues are clearly giving them setbacks, and now we have few rumours they are going to sell their few top players which are surely going to bring some problems into their consistency and quality but this all will be happened in next season.
Now most chances we will have them on second spot which could be also enough and Real Madrid is surely going to have good chance for staying at the top with they already maintain good lead, and they are also physiologically have good advantage from Barcelona after beating them few times in this season.