Break e2e test dependency on build artifacts

https://github.com/waltmayf/agentcore-amplify-fullstack/blob/5f2197550a7b02856c4cd01375905dfa7671398b/web/amplify/backend.ts#L198-L210

@claude, I would like to be able to run my end in tests, including playwright and others without first having done the build. I figured we could do this by uploading the relevant parameters to SSM parameters and be sure to use the branch name and the repo name in the to the perimeter in someway so we can distinguish builds on different branches in the end. I'd like to be able to in a fresh terminal where I didn't deploy the previously run all my end to end test including using playwright to test that the chat works. I'm totally happy if playwright is going to invoke the remotely deployed app the one that is deployed using cloud front backed by S3 for static website.

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*Imported from `waltmayf/agentcore-amplify-fullstack`#25 — originally filed by @waltmayf on 2026-07-01. Migrated to open-source repo; cross-references updated post-import.*

Start by reading web/amplify/backend.ts lines 198-210 and locating the Playwright end-to-end test setup. Define how the required parameters are stored and distinguished by branch and repository, then verify that a fresh terminal can run the end-to-end chat test against the deployed application without a prior build.

Written by the indexing model from the issue text.