I don't see any pros. Your liability for potential losses would not decrease simply because the card is linked to a different account.
I can see the reasoning of "if they don't have a lot of money in that account, the transaction will get declined and we don't take a loss". That may be true, but if the card number is compromised, the crooks will still find a way to use it. They look for patterns, they try cards on usual paydays when they expect there to be more money, etc.
PROS: none that I can see
* customer service issue
* MORE potential fee income losses, as customers would likely forget which card to use for which account and they end up overdrawing one by accident, etc. Due to Reg E changes, unless they opt in, you won't be able to charge them an OD fee.
* You're now expecting your customers to balance TWO checking accounts. We all know that people have problems balancing one.
* How are you going to enforce it?! You can't expect to call Mr. Customer and say "We noticed that an amazon.com charge posted to your account #123. All your online purchases should only be made to account #456." Really?