In answer to your questions:
1) Yes as a secondary employer it would be usual to see a tax code of BR (20%)
2) It is possible for an employee to choose who they want as their First employer and who should receive the Tax Allowance with. It is also possible to have their Tax code split between 2 employers. However this is the decision of the employee and you would not really be able to ask the employee to do this.
3) PAYE Tax is down to the employee to pay, so therefore only the employee would be able to claim back any overpayment in tax. The fact that your client has agreed to pay a Net figure to the employee does not matter, it is not an employers liability.
Your client I'm afraid will just have to bear the cost unless the employee is willing to change their tax code, however I suspect as they are earning a Net amount, and probably not paying any tax at their other job, they will not be very willing to do so.
Oh the joys of Net to Gross, I believe your client has just found out the biggest one, unfortunately to their cost.
Sorry I couldn't bring better news
Payroll Manager in the Healthcare Industry