There are various discussions here about how to use select the appropriate CLI profiles. I have not found any suitable solution until now.
In particular, AWS_PROFILE wasn't enough since cdk needs to make calls in different accounts for different stacks.

I wanted a solution that could:

1. reuse my existing profiles (~/.aws/config), but also
2. different profiles for different accounts (in my case, this implied different cdk stacks), while
3. keeping the cli invocation simple.

The trick is to use the plugin feature which really means just a javascript class.
You can define a class which will be loaded at runtime. It allows you to pick some profile for a certain account ID.

Here is the gist and an example cdk.json:
https://gist.github.com/jeshan/178dfa811df0f652b30d3cc61058512d/

Note that you can also specify plugins with: cdk --plugin ... deploy

I'm leaving this here as a how-to for the community. Feel free to copy the class into your project.

If anybody knows of a simpler way to achieve this, please let me know.