If you are working to ASME IX, take a look at Table QW442 for A numbers, both classifications fall under A No. 8, so you should be fine with one procedure. Depending on construction code, other rules might come in to play.
ER308LSi would be recommended for GMAW welding. As jwht stated, per ASME A No. is the essential varialbe. However, if impact testing is required, ASW Classification is a supplementary essential variable.
Do you mean that if impact tested is performed during WPS using ER308LSi, a job that required to use ER308L with impact tested can not use the WPS with impact tested using ER308LSi?
In answer to your last question, separate qualifications would be required for 308L vs 308LSi. The addition of silicon results in a new class of filler metal. I should have considered the Supplemental requirements with my initial answer, since I run into the impact situation regularly and have to justify re-qualifying to people who claim that, and other requirements can't possibly apply to us.