"currently being charged to Reimbursable Expenses account."
It's not, though. It's your Travel Expense. Look at your tax form. There is no line to report anything as vague as "Reimbursed Expenses." What you have is the initial Purchase entry as Travel Expense, and the payment to you is part of your sales income.
"The billable expense features of QBO Plus don't seem to be a good match for my requirements."
If you wanted to see and track it, you would set up a Service item or other charge, such as Travel, that you charge to them. What matters is that your own financial reporting shows Gross Travel as your expense and Payment from the customer as gross revenue.
"My client doesn't want to receive an invoice that lists all of the detail items of the expense report already in their system.'
Then don't use your own system for any details; but if you don't send that to them, then this is your own internal document for tracking AR and you get to put here what you want to track for your own purposes.
You either want to see it or not in your own system. You might not need an invoice at all in QB, if you log into their site to enter what they should pay you for. You would simply use Sales Receipt when they pay you, if you are not tracking the anticipation of their payment as an invoice for your own perspective.
You use the tools that matter. And you indicate you have Essentials. If you need more tracking, you should move to QBO Plus.