...i assume the PSD is CMYK, would be best if you can post a link to download the source PSD and indesign file, easier to problem solve when seeing what your dealing with...
...high quality setting includes tagged profiles, so it is likely you have a conversion happening on screen (although the color numbers are in fact the same as the source PSD), ensure your acrobat color management settings are the same as your other adobe applications, you don't need bridge to achieve synchronization...
...pdf x 1a is really the one you want to use at the very least, it does flatten transparency, so you have to ensure your artwork is layered correctly before flattening occurs (i.e. text on top of everything wherever possible). Any RGB colors will be converted during export (from indesigns RGB working space for native rgb objects, or from the RGB tagged profiles of imported RGB objects to the CMYK destination selected in the export dialog. The default destination will be your indesign CMYK working space. If indesign detects imported RGB objects without embedded RGB profiles, it will use the indesign RGB working space as the source...
andrew