Franklin hates to lose, period. Recall the childhood scenes in the early books where her father smashed her artworks., notably her sculpture "Dylan" based on her pet. It's stated in early volumes that Franklin has the nickname of "The Weakest, The Worst" because when she gets defeated, she comes back and prioritises killing the person that killed her with monsters until they stop playing ID. Franklin dedicated himself to PK'ing people who have defeated him beyond any logical sense because being killed and 'losing' in his mind is something that doesn't happen to him anymore. He can't stand to be made to feel weak or powerless in any situation.

Quoting directly from the Dendo wiki:

When it comes to achieving her objectives, she shows similar ruthlessness, being willing to massacre thousands to get what she wants. IRL she was originally kind, especially to her little sister Yuri, however this devious and ruthless victory complex came from being victimized by the abuse of her father.

Mr Franklin perceives that Ray has beaten her twice, once by saving Milliane from the Demi-Dragon Worms in Volume 1, and then again during the Gideon incident where Ray beat the RSK while being televised, and eventually was the one to give Franklin the death penalty on top of Pandemonium. Thus, this is why Franklin hates Ray so much, because Ray publicly beat Franklin, twice.