When you bring a higher than DVD resolution image into Encore, Encore downrezzes it to the project setting when you render the menu and/or build the project.
You can only avoid that if you make the background image the correct size. You can only avoid Encore rendering if you don't add any other elements other than highlights (the layers in photoshop with =1, =2 etc.).
If any of the images are part of a highlight layer, you get big problems; I don't think that is an issue for you.
Logos have often been scaled in various ways; I wonder if that could be having an effect.
But, I did not think the image looked that bad once it had it set to the DVD pixel size and was viewing it at the size.
You certainly do not get an accurate test viewing as your client was on a large computer monitor.
Create a test disk and view on an HD TV with an upscaling player. For me that is the modern, practical test.
The other test you might do is to take the logos out of the equation by making a menu with several fonts/sizes to simulate the text issue in the menu. DVD and small text are just not a winning combination. The logos may be scaled too small for so many on one screen.