Moving the culture slider is a last resort that you should try to avoid. You move it at the expense of your tech rate.
The one and only way and actually a nice feature of this scenario is to run for one or more wonders in each of your culture pressed cities. This applies when you play any of the European nations.
You would have to whip some workers and chop I think.
There are at least 5 wonders of interest available, Oracle / GLibrary / Mausoleum / Colossus / GLighthouse. (NB: HRE has copper and marble)
The first 2 would need to be done in 10T, the other 3 in 15-20T from start. Usually Elizabeth pops a GE T10 which takes GLib or Artemis.
For HRE probably Bremen can do Oracle and/or Colossus (if it has a forge already), Rome can do GLib and/or GLH/Colossus, Salzburg one of Oracle/Glib /Mausoleum (I don't think it has a library for GLib)
One wonder per city should do for culture fighting. You are also denying those wonders to your neighbours. Especially Paris, which has I think Notre Dame shouldn't get a 2nd wonder.
If I remember correct in the original scenario you start at war with Byzantium. But you start with enough units to take Cairo and then withstand counter attacks at any frontline right?