It is not officially supported. You can find the supported platforms at https://confluence.atlassian.com/bitbucketserver/supported-platforms-details-779362568.html
Having said that, if you run into an issue that is a genuine bug in the product, I am sure you will get support from Atlassian. You might have to reproduce the issue on a standard installation, if they demand it. And even on the docker end, you might get support from the community (if you are lucky enough that someone encountered and solved the same issue before).
I can't think of any reason why it wouldn't work with docker but you will have to test it yourself :)
Bitbucket Pipelines helps me manage and automate a number of serverless deployments to AWS Lambda and this is how I do it. I'm building Node.js Lambda functions using node-lambda ...
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