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supertest
🕷 Super-agent driven library for testing node.js HTTP servers using a fluent API. Maintained for @forwardemail, @ladjs, @spamscanner, @breejs, @cabinjs, and @lassjs.
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SurveyJS
Open-Source JSON Form Builder to Create Dynamic Forms Right in Your App. With SurveyJS form UI libraries, you can build and style forms in a fully-integrated drag & drop form builder, render them in your JS app, and store form submission data in any backend, inc. PHP, ASP.NET Core, and Node.js.
But of course you have to transpile the tests if you want to write them in Typescript. For Mocha you can use ts-node to compile them "on the fly" while running tests.
In your case it seems you do not need to include the integration tests in your image. You can just run the container and then fire the request against localhost and the port your container is bound to. For this I can recommend Supertest but your use case sounds like it could benefit from Pact too.