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.
This season it is possible for Madrid to get the trophy but when talking about the future it is a little further away although I agree when saying Madrid's current condition is much better but when talking about future opportunities everything will be the same when at the beginning of the season but for this season I am quite sure Madrid will be able to get the trophy back after previously being taken by Barcelona.
Some rotation will definitely happen in the end for next season and we certainly know with the current midfield conditions it is possible that next season Modric or even Kross may not be available in the Madrid midfield and it is possible that there must be an option even though currently their midfield is quite complex with young players but there must be other alternatives as well as for the Mbappe problem everything is still the same as usual where we just have to continue to wait for what will happen with the transfer saga next season whether indeed Mbappe will really be available at Madrid or there will be another drama like what happened in the past few seasons.