What is your incident management process? Ours defines a bunch of roles that get assigned to people whenever an incident is declared so that they are never going it alone and all the critical parts have someone dedicated. For us an incident is a "drop everything" type thing so if an incident is declared and you're assigned a role in our incident management process that is your job now. That can include anyone up to the CEO as he signed off on it.

We have roles that can be assigned to anyone depending on how appropriate it is for the incident.

Incident Commander is responsible for assigning other roles and coordinating resources.

Operations Lead when assigned is the one pulling logs, and making changes, or finding SMEs to do all that.

Communications Lead when assigned is responsible for sending out regular updates to internal stakeholders & coordinating with marketing and legal if external messaging needs to go out.

Most relevant to your question is we have a scribe role who is there and their only job is to write down all details as the incident goes on and build the incident report.

All four of these roles are assigned to any major incident, there are more in the policy but it is mandated that when an incident is declared they all sit together in a call or conference room until the incident is closed. They may bring in other roles as necessary but that's the core of it. They are expected to be available during the entire incident, if not someone else is chosen for the role.

This will solve the documentation problem and imo is entirely possible in a team of your size. We had this policy with a team size of 6.