I honestly very much doubt it.
First off, Spain will simply not let Catalonia gain independence. I don't believe the government of Spain is committing to what the UK government did with Scotland in terms of allowing and honoring a referendum that seeks secession.
Secondly, if Catalonia were to break away from Spain against Spain's will (unilaterally announcing independence without a deal with Madrid), I don't know how many countries, especially those in Europe will recognize it right away. Spain's neighbors such as France might not want to antagonize Spain. Ligue 1 might be reluctant to let us in because of pressure from Spain and other political risks.
Thirdly, even if we were to be allowed to join Ligue 1, there is no way in a short period of time there will be such a high-profiled rivalry like the one between us and Real Madrid that garners the same amount of interest it does today. After all, Real Madrid is billed "the most successful football club in the world in the 20th century" while we have been the most successful club in the last 15 years. Us and Real Madrid are the top two clubs in the world right now, these two clubs have the prestige, the money and the result to warrant their status. While PSG is rising fast with the oil money from Qatar, it is nowhere close to the overall status us and Real Madrid enjoy, not any time soon. Ligue 1 with us and PSG will not be the same as La Liga with us and Real Madrid.
Lastly, I heard the French are simply not as crazy about football as the Spanish, the British, the Germans, the Dutch and many other countries are. I read that rugby, not football, is the most favorite sport in France? If that is true, that just means that we will never get the same amount of money from TV broadcast we get from La Liga.
Us getting expelled from La Liga will be very, very bad for us indeed. Of course, Real Madrid will suffer as well once we are gone.