Air Carrier Captain reported a communication breakdown between the Captain and Dispatch resulted in an incomplete Flight Release due to missing onboard Hazmat cargo.

Narrative

We had an unscheduled aircraft swap in ZZZ. While walking down the jetway; which was full of boarding passengers; for the next aircraft; one of the guys from Ground Operations asked me if I could fill out the paperwork for a Dangerous Goods aircraft battery being transported. I filled out my section and took my copy into the cockpit. In the process of preparing for the next flight; I totally forgot that Dispatch needed to be notified and a notation put on the Release. During cruise to ZZZ1; the Dispatcher ACARS messaged us about the discrepancy.This was out of the normal sequence for any Dangerous Goods we have had in the past; where the Ops Agent will normally bring the paperwork to me and we will both make sure that Dispatch is notified and a note is on the Release. If this is the normal process; then we need to stick to that process to make sure that everyone who needs to be notified is notified. Additionally I need to be more aware that if something happens out of sequence; then I need to take some extra time to make sure that all of the I's are dotted and the T's are crossed.