ci(Tests): Migrate API Pull Request workflow to Depot CI

[khvn26]

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• I have added information to docs/ if required so people know about the feature.
• I have filled in the "Changes" section below.
• I have filled in the "How did you test this code" section below.

Changes

In this PR, we're testing waters for Depot CI by migrating the API PR workflow. Let's see how this works for us.

How did you test this code?

This is a CI change.

[claude]

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3 Skipped Deployments

Project	Deployment	Actions	Updated (UTC)
docs	Ignored		Mar 24, 2026 2:04pm
flagsmith-frontend-preview	Ignored		Mar 24, 2026 2:04pm
flagsmith-frontend-staging	Ignored		Mar 24, 2026 2:04pm

[github-actions]

Docker builds report

Image	Build Status	Security report
ghcr.io/flagsmith/flagsmith-e2e:pr-7030	Finished ✅	Skipped
ghcr.io/flagsmith/flagsmith-api-test:pr-7030	Finished ✅	Skipped
ghcr.io/flagsmith/flagsmith-frontend:pr-7030	Finished ✅	Results ✅
ghcr.io/flagsmith/flagsmith-api:pr-7030	Finished ✅	Results ✅
ghcr.io/flagsmith/flagsmith:pr-7030	Finished ✅	Results ✅
ghcr.io/flagsmith/flagsmith-private-cloud:pr-7030	Finished ✅	Results ✅

[github-actions]

Playwright Test Results (oss - depot-ubuntu-latest-16)

 10 passed

Details

 10 tests across 7 suites
 46.1 seconds
 b7e358b
 🔄 Run: #15447 (attempt 1)

Playwright Test Results (oss - depot-ubuntu-latest-arm-16)

 10 passed

Details

 10 tests across 7 suites
 12.6 seconds
 b7e358b
 🔄 Run: #15447 (attempt 1)

Playwright Test Results (private-cloud - depot-ubuntu-latest-arm-16)

 2 passed

Details

 2 tests across 2 suites
 1 minute, 4 seconds
 b7e358b
 🔄 Run: #15447 (attempt 1)

Playwright Test Results (private-cloud - depot-ubuntu-latest-16)

 1 passed

Details

 1 test across 1 suite
 50.9 seconds
 b7e358b
 🔄 Run: #15447 (attempt 1)