There really is no such thing as a Bitcoin "network" it is a loose organization of independent nodes.  A particular node has no way of knowing which tx is the original and which one is the modified one.  The modified one potentially could be received first.   There is also no way to reduce the compensation to miners as the tx and its duplicate are identical in terms of the inputs, outputs, and fee paid.

It may be possible to reduce the % of times that the duplicate gets included in a block by improving the tx selection criteria among miners who voluntarily upgrade their systems, but it probably would never reduce it to zero even among participating miners.  There is also the possibility that the attacker/griefer is also mining and favoring the duplicates over the original.   Both are valid copies of the transaction, a block containing either one is considered valid by other nodes.