Ask the agent questions about all the other teams' code, reaching out to them for questions it can't answer or clarification. With agent capabilities atm this is rare or can be done fairly async: "please confirm these things".
Maybe realise your code architecture is completely wrong. Manually code up some new abstractions that fit better, write the learnings into the spec plan. Strip out any implementation that largely doesn't fit your updated abstractions. Ask the agent to migrate the code to the new structure.
Repeat until spike is operational and you're happy with the abstractions used
Chat with the agent to create a Design Doc for the approach in the spike. Create a single JIRA ticket for "Productionise CodeShmode's spike". Get reviews and feedback from stakeholders.
Integrate feedback into your spike, or even the original spec document and regenerate the whole thing.
So much of the ritual you've outlined here is overhead from working in a large org where roles are siloed. When one person is empowered to do more then the actual work per person goes down and the overhead becomes the dominant. But that overhead isn't needed anymore because one person can now do many people's work.
I've whipped up spikes in a few days that would've been a month of work across a team multiple DDs and approvals. In the past this wasn't feasible so we would need to justify what those people would work on. Now you can whip it up, show a working demo and ask "should we productionise this"