You're not even able to do your politics effectively without a middleman that can easily undermine you.
The only path to career advancement to this is to job hop as much as possible.
I'll tell you my own personal experience here: Go talk about it!
Depending on the culture and rituals at your company:
- Ask for a spot at an all-hands meeting so your team can showcase what you shipped
- Record a demo video of the team's work and share it in Slack/Teams/Yammer
- Write a post-mortem or internal blog post about the impact that your "ship" had on the business.
- Etc...
I'm a firm believer in the concept of "internal marketing" in big companies. I've had leaders and mentors repeatedly tell me "no one will recognize you in a big company if you just sit around waiting for the recognition"...
You have to put it out there and talk about it and show it off...
And yeah, I know a lot of tech folks hate doing this.. But in my experience that's how you play the game in an honest way... Find the right time and place to be proud of your work and show it off.