"I've downloaded a custom template for Invoices from QB online."

From the Gallery? Not from the QB Online Program?

"There is a field in the heading for Project name."

That is Job. Open Customer Center and Add Job = subcustomer, for tracking.

"The field is not typically wide enough for the data that I have entered for my projects."

Job Name is limited to 41 characters, I think:

Character limitations for fields in QuickBooks

"The data is truncated, with "..." appended to the end."

You Customize the design using Customization, then Layout Designer, to get to the Graphic View, to change graphic elements and widen that field.

" would like this field to wrap to the next line"

It doesn't Wrap. Go to Layout Design and move that, for instance, to above the body color block on the right side and below PO#, to make room for you to make it Longer. I don't like the text fields mixed into the accounting area with due dates, anyway. Or, shorten the Job Name, such as WRV for Wood River Valley. Or, move that entire block of text to the left. Move "Project" label box to left align with PO, and make the Data Field box wider. Or, drop the font size just a bit.