Can you provide more information, for example the REST API calls or resources you called.  In Test Lab you need to create test instances in a test set, in a test folder, etc.

Check with your project admin and see if there is any configurations such as data hiding. Ask for help from DBA and see if the test cases can be seen in the database. 

What's the type of your test lab?

If it's functional, only some type of the tests can be selected: BPT, Flow, QuickTest, Service Test, System Test, VAPI-XP Test, etc.

If it's performance, only load runner related tests can be selected.

If it's default, please provide more info about the test and the test lab.

As per your description, it seems your ALM project has been customized. You said "they are approved in Test Plan", by default, ALM does not has the restriction to approve a test case before execution, please work with your process/project admin to check if the data you created through rest api compliant to the project regulation which is defined and customized by project admin or your team expert. 

You are also welcome to log a support ticket and ask them to elevate the case to R&D. We will need to review your project configuration together in order to resolve the problem. 

The user posted this question 2 months ago and has not replied to any of the suggestions, so who knows what's going on.

It's also entirely possible that they created test cases in test plan but didn't actually create any test sets or instances in test lab.  Which would also explain why they say they can't see the tests in test lab.

But without the user giving any response or follow-up, there's no way to help unfortunately.

It doesn't matter, maybe he/she already got the answer. The question and answers are also valuable for other people who may meet the same challenges.