Relative Priority

Our customer has a system to track all problems that their users report. One property of trouble tickets is their priority. This priority ranges from 1 to 5 (1 being highest) based on a strict set of criteria. We are nearing the end of our software maintenance contract on our project. Most of the problems that get reported these days are priority 4.
Recently our customer headquarters group has been thinking about raising the priority of some problems which are crucial for their success. Their goal is to get the trouble tickets fixed quickly. However this will only lead to trouble. High priority tickets require a lot of status reporting overhead. So upping the priority might actually delay the resolution of certain problems.
I think the root cause of the problem is that there is no way for the customer to indicate the relative priority of problems they report. Our process is to work the issues in priority order. And if two or more tickets have the same priority, we work on the oldest ones first. Another part of the problem is that our software development team is shrinking in size due to the end of the contract. Us remaining developers are doing the work of two people these days.
My response to the customer concerns was to have the developers drop whatever they were working on to concentrate on the few high priority issues. It is my sincere hope that we can knock out these problems and leave the customer in good shape. This will also make my life easier as we try to close out this maintenance project.