Add text layer to existing print.

Sounds like a basic physics issue...

https://en.wikipedia.org/wiki/Subtractive_color#CMY_and_CMYK_color_models_and_printing_processes

If overprinting text as a second pass on a previously printed inkjet print, my expectation would be to use black and understand that the image may show through in some areas as density may not be high enough for large/thick type or panels.

This is after all a compromise that you have chosen, rather than reprinting the images again with the text combined in a single pass print.

I am presuming that this is an aqueous-based inkjet and not a UV or latex/resin printer or eco-solvent printer capable of printing white ink.

Edit: I would have informed the customer that you had already printed to meet their delivery expectations and that they would be charged again full price for new prints, however, they could take the first prints if they wished as they had already paid for them. Or you eat the cost of a free reprint... Or you get what you get with your overprint.

Transparency is the absense of any color. It contains no white ink.

Can you post a screenshot of your Layers Panel showing us the type layer?