The problem would be when to get people from the West End in. They usually do 2 shows on a Saturday; the evening one would clash with the recording of SCD in both the live and "live" capacities. And then if they were to pre-record before the live show, the audience wouldn't in attendance because they're still queuing/in the holding room.
But I agree, I would love to see some West End stars performing, the only way it'd work is if the BBC used the standby's/understudies instead of the leading cast members.