I think Champions League is biggest reason they try to keep Mbappe. With great vision and key passes from Messi, Mbappe will have more opportunities to show his speed and score more goals. The combination will increase chances to win Champions League for PSG.
They might also be confident that they can convince Mbappe after all, to renew a contract.
Messi arrival to PSG will actually increase the prestige of the French league and in my opinion, if Mbappe continue to stay at PSG, of course it will reduce his chances of winning the Ballon d'Or and of course Messi big name will make mbappe lose his prestige at PSG, I think more better for mbappe to go to real madrid instead of having to stay at PSG at this time.
That's one of the most plausible reasons at the moment about why PSG didn't sell Mbappe this season to Real Madrid. There is no other big reason PSG signed Messi from Barcelona other than improving the status of the French League 1 for the better and giving them the opportunity to win the champions league title this season or next season. At least not a big problem for PSG so far rejecting Real Madrid's offer in my opinion.
The most realistic reason PSG brought in Messi is to get the champions league trophy, because so far PSG have always failed to get the champions league trophy even though they have a squad that is quite capable and always qualify for the champions league.
For the Ligue 1 Cup, I think the presence or absence of Messi at PSG is certainly not very important, because PSG have dominated Ligue 1 so far.