The ethics of team morale... what would you do?

As the manager, I would ask myself do I really know him? There may be factors impacting his work behaviour that stem from his home life. Therefore, before I decide to undergo the performance management process, I would like to get to know him better and understand his mindset. If you decide that his mindset is fixed, then redundancy may be best suited, thus allowing you to allocate your time more efficiently.
Also, as he works in a customer-centric industry, there is no excuse to be consistently rude to customers. If he isn't open to improving and developing then you need to start the separation process as soon as possible.

From the outset, it is clear to see that there are a few behavioural issues at play here;
1) Team member is consistently rude to customers
2) The team member's rationalization of his unacceptable behaviour (feels entitled to be rude as he sees more customers)
3) The team member's unwillingness to take on feedback or coaching
4) Overall lack of respect for others (towards customers, the team, and the manager alike)
Since this is causing problems with Customer Experience, team culture and the team's morale, the manager's decision to initiate a formal performance management process is absolutely what is required here - but will need to be done so with care.
Specifically, the Manager will need to take care in understanding the root cause of these behaviours, as there may be more than meets the eye with the Team Member's personal circumstances (i.e. Relationship breakdowns, financial hardship, mental health issues etc).
Additionally, it might be worthwhile to review/re-assess the team member's workload, as the team member is clearly stressed and feels that "he sees more customers in a day than any other team member" (which very well may be the case).
The "Easy way out" for the manager here would be to offer the team member one of the redundancy packages - but the right thing to do is to take the time to understand the root cause, and then to support the team member and give him every opportunity to improve, and get back on track.

From the outset, it is clear to see that there are a few behavioural issues at play here;
1) Team member is consistently rude to customers
2) The team member's rationalization of his unacceptable behaviour (feels entitled to be rude as he sees more customers)
3) The team member's unwillingness to take on feedback or coaching
4) Overall lack of respect for others (towards customers, the team, and the manager alike)
Since this is causing problems with Customer Experience, team culture and the team's morale, the manager's decision to initiate a formal performance management process is absolutely what is required here - but will need to be done so with care.
Specifically, the Manager will need to take care in understanding the root cause of these behaviours, as there may be more than meets the eye with the Team Member's personal circumstances (i.e. Relationship breakdowns, financial hardship, mental health issues etc).
Additionally, it might be worthwhile to review/re-assess the team member's workload, as the team member is clearly stressed and feels that "he sees more customers in a day than any other team member" (which very well may be the case).
The "Easy way out" for the manager here would be to offer the team member one of the redundancy packages - but the right thing to do is to take the time to understand the root cause, and then to support the team member and give him every opportunity to improve, and get back on track.