Use a separate amp for music is definitively an alternative but the connection can be very troublesome, especially when you have to share the same pair of speaker!
That is difference when you select the effect from the BDP, in such case, you are using the BDP built-in decoder.
Are u using the HDMI cable for to connect the BDP and your amp? If yes, then make sure your BDP is set to output bitstream signal via the HDMI cable.
And I believe yur amp when receive the DTS and or DD signal it can recognize it and decode it! You should not need to select the (movie/music/game) selector, those are for 2 channel stereo signal.
You were simply playing different sound channel in the BD. nothing related to decoding.
Your amp should recognize the encoding method of the signal anyway.
If you sure you have select the DTS sound track at eh BDP and cannot get it decoded by your Amp. You may want to re-study the manual of the Amp. and make sure if there are any connection or settings are missed!