So, I want a sub that plays loud even in low volume.
This statement confuses me.

Do you mean to say rest of the sound at low volume and only sub playing loud?
That is likely to make sound muddy.
Please have a home audition if possible before putting down your money.
Given your taste of music,I think you need a balanced presentation of music which your present speakers are not doing. Classical and gazals have so much importance in mid range is that myself and none of my friends bother much about the bass when we listen to this type of music.
I do agree that bass guitar is prominent in some gazals( I have noted that mainly in the newer gazals by late Jagjit singh)
If you dont mind,may I suggest something? Before buying a sub, please have an audition with a pair of good bookshelf speakers coupled with your sony system
(provided the sony has non proprietary speaker connectors). In your budget some good bookshelf speakers are available.
Integration of sub in stereo music ('invisibility' that you have mentioned) can be a headache unless you have active crossovers or variable low pass filter to try out various cut off frequencies.
Many of the times,stereo music is produced better by good stereo speakers than a sub paired with poorly made speakers.
I hope you dont mind for this off the track suggestion.