"However, there is one set of features that we need and can't find: we need to create estimates (base bids), enter change orders if necessary, and invoice a combination of base bids and change orders.  We also invoice for parts of the project complete (I guess this is called progressive invoicing).  On that basis, we then need to generate Customer Statements for jobs."

Which is why you would use Desktop, not Online. You have a sophisticated need, but you seem to consider that Not Complex?

If Premier is too expensive for you, so is QB Online.

What you are asking is mostly available in QB Pro. You can Downgrade that data file into Pro, and there are Work Arounds for Charge Orders; you add a new Job level for that CO and its own estimate, for instance.

Pro handles ProgressInvoicing and Statements and Estimates just fine. You will have a few missing Job reports, and no Costs to Complete reporting.