QB Issue Resolution:

to answer your question,

permission to write checks, but not print them.

Not an option, if they can write checks then they can print them

Nothing in QBO permissions addresses payroll access or functions, so I can not help with that.

Resolution for Issue 'I would like to give my employee permission to write checks, but not print them. I also want them to update accounts and create reports, but not do payroll.' available: Yes (Solved).
Source: Intuit Community forum.

I would like to give my employee permission to write checks, but not print them. I also want them to update accounts and create reports, but not do payroll.: this issue or error code is a known issue in Quickbooks Online (QBO) and/or Quickbooks. Support for this issue is available either by self-service or paid support options. Experts are available to resolve your Quickbooks issue to ensure minimal downtime and continue running your business. First try to resolve the issue yourself by looking for a resolution described below. If it is a complex issue or you are unable to solve the issue, you may contact us by clicking here or by using other support options.

Support for addresses payroll
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Error Recovery for addresses payroll [Canada]
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