How to solve Bus Contention problem in my case?
This is not bus contention - there is no bus.  RS-232 simply doesn't work the way you want it to.  Period.  End of story.  It's point-to-point and not a bus.  I've seen serial cards in laptops get destroyed because of 'clever' wiring schemes.

You talk about applying an undefined level to it and ask how you can do that.  It also can't be done.  It makes no sense to produce an 'undefined' voltage - when the electrons hit the other end the voltage is defined, period.  You can't send an undefined voltage level to it. 

You'll probably reply with a comment to this post asking if this response is for real.  It is.  You can do nothing to solve this without putting some sort of arbitrator where the two red wires meet.  And as you said - that involves new hardware.  So game over.

RS-232 simply does not work this way and you will probably damage things trying to make it work.