Advanced Party Sheets can only be taken when all members of the party are at least Rank 2, but do they have to spend experience to switch from a Basic Party Sheet to an Advanced Party Sheet?

No. There is no XP cost in changing party sheet, either from basic to advanced, or between two basic, or from an advanced to a basic.

The GM's Toolkit suggests that the players should be able to change party sheets for free at "key points" during their career.

I believe rules also state (GM's or base don't recall) that at other times it costs 1 advance from each Player to shift.

I would allow free shift whenever roleplaying justifies it (mechanical card follows roleplaying). You've been intrepid explorers but been taking lots of risks, okay brash young fools it is, no argument, etc.

If I'm remembering aright, the "1 advance from each player" cost is if the group wants to acquire a second party sheet while retaining the first; it's an option mentioned in the GM's Toolkit . Just changing from one party sheet to another doesn't seem to have this cost.