According to the result of a query in Expertflyer 9W *can* check baggage through on to CX. However if they are willing to do so on two separate tix -I don't know.
depending on your nationality and the time between the two flights - isn't there an option to do Visa on arrival in DEL or has this been replaced by the eVisa scheme that the OP doesn't have enough time for?
good luck!