Free Checked Bag with Delta Gold Amex When Booking Through Chase Travel?

I'm booking an international trip on Delta: outbound on Delta from my city, then Delta codeshares operated by Air France/KLM, with the same routing on the return. Basic Economy. No medallion status on DL.

I plan to book through the Chase Travel portal because I have a $500 credit there expiring soon. For the remaining $, I can either use my Chase Sapphire Reserve (ideal) or my Delta SkyMiles Gold AmEx Card (if necessary.)

My question: will I still receive the first checked bag free benefit from the Delta Gold Amex?

From what I understand, this benefit is tied to my card membership and SkyMiles account, not the payment method or booking channel. As long as my SkyMiles number is attached to the reservation, Delta's system should recognize me as a Gold Amex cardholder and automatically waive the first checked bag fee. It shouldn't matter that part of the ticket is paid with Chase Travel credit and the rest with my Delta Gold Amex or Chase Sapphire Reserve. What does matter is ensuring my SkyMiles number is linked to the booking (and I know I can add it later in Delta's "My Trips" if Chase Travel doesn't capture it.)

Am I correct in this understanding? Please confirm. Thank you.