You receive a check for payment for a fairly large value issued by the biggest customer of the agency, only the account presented insufficient funds, impossible to reach the customer. What do you do in that case?

Please correct as Cheque and not check.  If you have power and limits with the manager and turn over and relationship with your customer is excellent and you are sure that previously made payment and you are risk taker for retaining such  high customer.  You can clear the payment despite the fact it is an operational violation.  You must be in a position to clarify your stand otherwise you loose your job instantly.

If you are Empowered to overdraw the account taking into the Relationship size and the profatibility of the account the same can be honoured, but if you do not have powers then there is no option but to return the cheque without putting yourself at risk.

In that case you would communicate with customer in writing that is via email mainly explaining that such a situation has occured and a reply as soon as possible would be appreciated.

This case should also be reported to your boss to who you report to and this communication should also be done in writing.

In case of receiving a reply seems remote, travelling to the customer's premises/office yourself and trying to get the matter solved, provided you have explicit authority to do so by your boss

The answer to this question varies as per the current standing(net worth) of the customer with the bank and adequate due diligence for cheque being issued by the customer. If both, i.e. the total net worth of the customer is more then the amount of the cheque and also if in the due diligence process it found true that the cheque has been issued by the customer. Then we can go ahead and take the call for paying the cheque and putting some sort of lien/mark on his other account. However, this issue has to be resolved as soon as the customer gets back.  

As a Banker, you need to do all means to advice the valued customer to have their Balance checked.  And to get proper instruction to fund the incoming check presented for clearing.  You do not want to return the check due to DAIF.  It will damage their reputation.  The best solution is to have all contacts with their Finance, Treasury or their Management. Not informing all of them on the scenario but getting the right person to act quickly to solve the issue.