First of all, if you write your tests to cover "required level of tests" or requirement to "have some tests at all" having the production implementation already done, it is slightly too late. In the majority of cases having tests first, based on your requirements, contract, use case or anything it more optimal approach. Nevertheless, I don't know your situation and the thing you're trying to implement, so treat it as a suggestion and move on to the key thing you are asking about.
Your JUnit (preferably 5) and Mockito tests, which probably use MockMvc are very good unit(-like) tests to cover web communication concerns such as: HTTP request types, content type, encoding, input and output parameters JSON (de)serialization, error handling, etc. They are best to test with the service layer mocked. Thanks to that you can easily cover a lot of web cases without the need to prepare data in a database, etc.
The core logic has to also be tested. Depending what it is, it might be feasible to test it in the unit way (the easiest to write, can cover a lot of - also corner - cases). It could be supplemented with some set of integration tests to verify that it works fine also in integration (Spring Beans, DB, etc.).
If desired, you may also write some E2E test from the web call via (real) HTTP requests through controllers, services to a database/datastore (if any), but I would limit it only to the most important scenarios to use it in your CI/CD pipeline to verify that a deployment finished successfully.
Disclaimer. I've found this approach useful in multiple situations, but definitely in some other circumstances it might be good to change the balance point to better apply testing.