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For this season, Barcelona will try to catch up even though it is very difficult because they are already far behind Real Madrid who are at the top of the standings, but with their quality which is now getting better I think they will try as hard as possible because mathematically they still have a chance to do that. .
And realistically Barcelona only has a bigger chance in the UCL so now they are also shifting their focus there to give their best performance, now Barcelona is in the quarter-finals so their journey is not long, even though it is difficult.
Barcelona is almost certain to lose the La Liga title this season when Real Madrid leads the standings with an 8-point advantage with 9 games remaining. Real Madrid is comfortable enough to maintain its performance until the end of the season, but if they drop a few points in the next few games then maybe Barcelona can get closer.
The matches against Bilbao and Mallorca are something that Real Madrid must win before El Clasico, so if both can be won then it doesn't matter if Real Madrid loses in El Clasico. However, it would certainly be dangerous for Real Madrid if they lost one of the matches against Bilbao and Mallorca, they would be in danger of losing points in El Clasico too.
You are correct in saying that Real Madrid needs to win its matches against Athletic Bilbao and Mallorca. If Real Madrid wins these games, it will be extremely difficult for Barcelona to defeat Real Madrid later in the season. Even if Real Madrid loses their next El Classico match against Barcelona, they will still lead the team with points. Those matches between Real Madrid and Athletic Bilbao and Mallorca will be extremely difficult, but from the looks of it, Real Madrid knows how to play those teams very well and how to win them, so I have faith that Real Madrid will prevail.
Real Madrid must exercise caution in their match against Bilbao, though, as Vinicius will not play due to his suspension for receiving yellow cards in the previous game. Instead, Rodrigo will play the position, and Brahim Diaz will play on the right wing, ensuring a potent offensive. If Real Madrid plays lightly against Bilbao, they will not be happy with the result, as the team is dangerous these days, particularly with regard to the two Williams brothers who must be well-marked in order to deny them a chance to score.