SGS is used for slayer due to its special attack (hp and prayer restore). It fills the same purpose as Guthans for healing but has a guaranteed heal and less switches are required, which could be important if you need a slayer helm for your task.
Definitely the sgs. You can lazily flick piety and just restore the prayer you lose with sgs. It also heals you and you can spec some big numbers since you can keep the slayer helm on.
The saradomin godsword definitely offers the best sustain. The cost itself shouldn't matter much because you can just sell it off when you're done training slayer.
Buy sgs however if its costly you should buy guthans as it is fairly more cheaper and get the job done also the sgs heals / gives you some prayer back so its better also id rather use something like a dbattleaxe + restore pot for the free boost and bring prayer if you cant afford guthans