"I need a report that says how much I have billed against a job before sending in an invoice."

You selected that you have an Older program, and we don't know which, Pro or Premier.

Open the Income tracker, if you have it, and you see a listing of Billable time and costs. Also, other than Pro, you can enable this for Invoice From a Table of Time & Costs, in Preferences.

"The closest I have found is a report, "Company & Financial / Income by Customer Summary." "

There reports to examine Sales to date, but not the value of Pending to be Charged from Actual Time listed as tasks on Timesheets.

You can compare Qty and Item, such as Scope of work entered on an Estimate, to Timesheets: Time by Job by Item, or Time by Name by Item by Job, comparing Est qty column to Billed, Unbilled and Not Billable.

"However, it has a notable drawback. It always shows every customer/job I have had over the time period I select, both active and inactive."

In this example "Active" = Has Values in the timeframe. That is not a Status, in this example.

"I have lots of inactive customers/jobs. The report will not remove inactive jobs, and hence is 3 times as long as it needs to be and is difficult to work with."

That isn't the right report for what you are asking, anyway.

You want Job Profitability or Sales reports. Or, Estimates Vs Actuals.

"I have many customers/jobs and manually using the filter screens to select the customers/jobs that are active will be quite tedious and error prone. Is there any alternate approach to get a report that only shows all active jobs over the time frame I select?"

Are you using Job Status? If your completed jobs have Status = Complete or Closed, and In Progress have status = In Progress, you can filter on one selection = Job Status.