You would need to create an automation (converted into a shortcut, as you mention) that runs every Saturday at 10 PM and test the week number to see if it's even or odd.
Here is an example:

- The 1st action takes the current date and format it into a week number.
- The next action divides by 2 and return the remainder.
- The "If" action tests whether the remainder is 0 (even week number). Set it to 1 for odd week numbers.
- Inside the "If" action, add the action to turn the light on.
So the automation would run every week but the light would turn on only on even (or odd) weeks.
Note: When you select the weekly repeat in the automation, unselect all days except Saturday.