If your new job is a similar one, you can just go ahead

to switch.Then you ask the new company to reply the CIS' requirement and state that it is a similar job as the one in your LC (or in the same field for NIW, EB11 case). I don't think it will cause any problem with your GC. Your first choice will make GC straight but may cast some shadow in the future. If you switch job shortly after GC's approval,your intent may be questioned later. This is a gray area.