Seeing how the LCY flight is all C-class - you have some leverage as a high-value customer.

BA has been tough on baggage if you originate your trip with them - even for their own employees who are interlining baggage.

The opposite though, has been very successful - starting a trip on another carrier (FB, LH, MA) and interlining onto BA

Hope for the best - ask DL to check it through with the right flight info. At BA check-in and gate, ensure that they scan the bag tag or manually link it to your PNR, if it is not linked already.

Hope this helps!