Our development environment is in the same situation. We're developing a solution containing web api's (micro services) which have an Angular front-end. For testing, we unit test the API's using SpecFlow and nUnit. We write scenarios which we test by actually starting the API's and posting models (DTO's) back and forth between the API's and test if the outcome is as expected.
On top of the, we use the exact same SpecFlow scenarios to start selenium tests for the front-end. We don't include all tests in these test, but the most important scenarios will be included in these tests as well. This way we make sure that out front-end is still capable of performing the most important tasks for the end-user in each and every version delivered by the team. Front-end tests are pretty expensive to develop / maintain, which is why we don't include all unit test scenarios in the front-end tests as well.
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This comes close to the solution we use right now for API testing :
End to End Testing of Web API 2 using OWIN and Specflow | Mohammad Ruhul Amin[
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