Ah the saddle is very shiny as are the leathers, I'll try some balm on them thank youShiny surfaces squeak. This may not help you but I take my own leathers with me when I ride (plus my safety stirrups ) and put them on any horse I hack. My leathers have never squeaked. But they are not shiny.
I clean my leathers with Effax Leder Balsam. The leathers are Amerigo and very soft. I take off the irons and keep the leathers hanging up behind the cellar door, so the leather remain straight and soft. They are almost 20 years old but they do not shine.
Well I bought it new but half price so maybe this is my punishment for it being a bargain"Old Timers" used to say if "your saddle squeaked",you hadn't finished paying for it.