Currently, in order for SGS to support the Black Eels the only way to do it would be to pass a 30 persuasion skill check. Unless you're already heavily investing into persuasion, there isn't a reason to dip 30 in persuasion just for a slight change in end slides.
This is fine for pre-drill as the PC is a relatively new face and has now gotten embroiled in a gang war, but it doesn't make sense for post-drill/post-Faceless invasion. By that point the PC should be a more known local figure and the potential gains from cooperating with a strong southern naval power should allow you to convince the SGS leadership to support the Black Eels without a need for persuasion.
Hell it doesn't necessarily have to be right after Faceless invasion, there just should be a point where the benefits and sway of supporting the Black Eels outweigh the negatives such that there isn't a need for a persuasion check. This could be after completing the Missing Merchants quest for Vera and SGS determining that Railways might be dangerous enough to consider allying with a powerful naval option.