I believe the reason why Mr. Bennet is willing to let Mr. Darcy handle this is because, he can very well afford it. Mr. Gardiner would not have been able to, which is what worried Mr. Bennet so much. The bailout would not have taken place without Darcy. $10,000.00 was Mr. Bennet's estimate, and he couldn't imagine how Mr. Gardiner could afford that. And further more, Mr. Gardiner had absolutely nothing to do with Wickham, (despite Mrs. Bennet's assertion that why should he not pay for his neice? Mrs. Bennet might have pressured Mr. Bennet not to pay back her brother, since she is the one who doesn't know how to economize, according Jane Austen herself.)

Mr. Darcy felt the obligation himself, and if Lizzy told the circumstances to Mr. Bennet, and since Mr. Darcy is not one whom any refuses, Mr. Bennet is happy to oblige, since settling it was also a matter of honor to Mr. Darcy as well.