ci: Add flagsmith-api-private-test image build

[khvn26]

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Changes

Adds a test image that includes private dependencies to be used for development and integration testing.

How did you test this code?

Added a trigger to build from this branch.

flagsmith-auth-controller integration tests run successfully on top of this image, see https://github.com/Flagsmith/flagsmith-auth-controller/pull/33.

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3 Skipped Deployments

Project	Deployment	Review	Updated (UTC)
docs	Ignored		Jan 16, 2026 8:49pm
flagsmith-frontend-preview	Ignored		Jan 16, 2026 8:49pm
flagsmith-frontend-staging	Ignored		Jan 16, 2026 8:49pm

[github-actions]

Docker builds report

Image	Build Status	Security report
ghcr.io/flagsmith/flagsmith-e2e:pr-6543	Finished ✅	Skipped
ghcr.io/flagsmith/flagsmith-api-test:pr-6543	Finished ✅	Skipped
ghcr.io/flagsmith/flagsmith-frontend:pr-6543	Finished ✅	Results ✅
ghcr.io/flagsmith/flagsmith:pr-6543	Finished ✅	Results ✅
ghcr.io/flagsmith/flagsmith-private-cloud:pr-6543	Finished ✅	Results ✅

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 98.09%. Comparing base (4ce72ad) to head (88ab8ce).
⚠️ Report is 1 commits behind head on main.

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #6543   +/-   ##
=======================================
  Coverage   98.09%   98.09%           
=======================================
  Files        1293     1293           
  Lines       46583    46583           
=======================================
  Hits        45695    45695           
  Misses        888      888           

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