There are quite a few answers here that miss the mark.
The client I am dealing with isn't a healthy customer, making demands at 1AM (their morning) and expecting one person to do the work of 10 people.
You'll find the client doesn't really care about how many employees your company has, or how many people are on support. What they do care about is getting the level of support they are accustomed to, or the level of support they are paying for. Their expectations are not personal, so don't view them as such.
If you do find yourself on support, you don't simply get to decide to deprioritise the work. You can't ask the team lead to talk to them. It's really outside of their control unless team leaders typically negotiate contracts. For all you know, they may be paying a pretty penny for platinum level support, and all they get is a single overworked resource, so you can't just go to them and say: "Sorry, I will answer at 9am my time".
You need to act in accordance with what you and your boss agree on, and this should be in alignment with customer expectations.
What if the company is not able to find people to do support? Then it becomes a business decision. There are several options the company can go down. These include:
- Renegotiate the contract with the client
- Terminate the contract with the client
- Wear the financial penalty for missed SLAs
- Hire more people that can conduct support
- Outsource the support
- Shift the focus to bug-fixing to reduce occurrences of invoked on-call
- Restructure support to be tiered
How can I refuse a team lead position that would entail working with an unreasonable client?
This is your actual question, and I've spend my time so far discussing other answers.
You are in control of some things. Firstly, it is within your capacity to talk to your boss and indicate the types of work you would rather not do. Would you do it for more money? What if was just during the week? Maybe you would settle to be part of a rotating support roster? There is almost certainly some level of on-call you are willing to do for the right enticement.
In addition, there are laws and guidelines in several jurisdictions that limit the time and scope of on-call. You should see if there are any in your location. I believe it's unethical for someone to be on call for 4380 hours straight.
It would be excellent if you could find reasons why you can't do support, or why you would be limited. Do you play sport? Do you have poor phone reception? Poor internet connectivity? Have to drive the kids to school in the morning?
In a perfect world, you don't need reasons, but it would be less of a career limiting move if it seemed like there were underlying reasons why you are unable to do support.
In the short term, you should try to learn as much as possible from previous projects and how they could have handled support better. Is your overseas support team well-equipped to handle the bulk of issues? Do they have mitigative strategies for when the client reports issues? Can they clearly identify severe and non-severe issues? Is there capacity for the client to fix things when they go wrong without involving support?