Not sure if this helps, I've no comparison to your workflow - or what your workflow is?
But are you adding the lettering on a layer on top of the photoshop image?
What file format are you saving from illustrator? PDF, AI with PDF compatible file, other?
How are you importing the image into InDesign? Copy Paste, File Place, other?
What happens if you just place the Photoshop image in InDesign - then File Place the lettering from Illustrator into InDesign over the Photoshop image?
What are your pdf output settings from InDesign?
Any screen shots of your settings when saving, importing and exporting files would be very helpful.
I'm lettering in a layer (two layers - one for balloons, one for text) above the layer the image is placed in.
I'm saving out of Illustrator as a PDF compatible Illustrator CS6 document - but I have tried various legacy Illustrator file formats, with options checked and unchecked. No change.
If I place the PSD then the Illustrator lettering over the top, the image renders correctly - but based on current behaviour, I would expect it to.
It doesn't matter which PDF export options I run, I get the same result with each.
I just ran a test, as I have RGB and CMYK versions of these pages, for web and print (in the CYMK version, both the placed PSD file and the AI document have been converted to a FOGRA39 colour profile) - the RGB page rendered correctly as a PDF. I don't know why I never looked at this before but I wouldn't have tried if you hadn't replied, Eugene - thanks.
Are the .psd files linked or embedded in the .ai? Does it work better if you save from Illustrator as PDF?
Is ID involved here because you have multiple pages, or some other reason?
Add the text in Photoshop. Save as PDP (Photoshop PDF) and place that in ID.
Very important to have the Text as a layer - don't rasterise it or flatten it with the image.