QB Issue Resolution:

Menu Edit>preferences>time & expenses> Company preferences Tab check the mark all expenses as billable

 

Then when you pay a bill, there will a block for customer name and a check mark box for billable.  Select the customer name and do NOT mark it billable, that associates the expense with the customer/job.  Using a job for this allows you to run a P&L by job from the job screen

Resolution for Issue 'Question: How to link a vendor bill with an invoice without billing it to the customer?' available: Yes (Solved).
Source: Intuit Community forum.

Question: How to link a vendor bill with an invoice without billing it to the customer?: 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.

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