atw: Pairing commas around a word or phrase

"The waiter was in the process of making another, conflicting booking." 

In this sort of situation I confess I've never been sure whether another comma 
is needed after 'conflicting'. If this were the latest in a series of 
conflicting bookings, no commas would be needed (though it would be time to 
look for a new waiter). That there was another booking is not necessarily a 
problem. The fact that it conflicted with another booking *is* a problem. 

The Chicago Manual of Style is the applicable reference for this project. I've 
been looking in CMOS but I'm not sure which guideline applies. I don't think 
'conflicting' in this case is appositional or parenthetical--is it a coordinate 
adjective? 

If this usage is different in the UK and Aus I'm interested to know that too. 

Thanks, 
Stuart

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