As of now, you cannot send individual expense entries from Core to QuickBooks Online; you can only get them from there into Core. So when using both applications, you should enter all your A/P transactions in QuickBooks (say bills, credit card expenses, cash expenses, checks, billable and non-billable expenses) and then sync them into Core. Then invoice the billable expenses you got from QuickBooks in Core, provided you had entered them under the correct Core project there. Note that the projects from Core are send to QuickBooks as sub-customers, so you should enter the billable expenses against the relevant sub-customer.
Another workaround if you have limited expense transactions is to enter the expenses as vendor bill items in Core. The vendor bills are synced both ways so you do not need to enter the expenses in QuickBooks. They are sent over when you send the vendor bills from Core to QuickBooks. However, this is limited to vendor expenses only. For employee expenses, you need to enter them in QuickBooks Online and then get them into Core as you cannot send them to QuickBooks. Check the Core Help for details on Core-QB Online data mappings.