Accredited status means that you have the status of being accredited. Accreditation status refers to your status of being accredited or not. Thus someone might ask what your accreditation status is, and you would answer that you were accredited or not. They might alternatively ask if you have accredited status, and you answer yes or no.
(That refers to the case where it is a binary batter, yes or no. There is also the less common possibility where some category of entities, such as people or companies, has some sort of status which has several possibilities, and that status might be accredited or not. That is not the usual expectation when those words are used, though.)
A testing laboratory is a laboratory where testing goes on. A test laboratory might be a laboratory that conducts tests, but might also be a laboratory that is being run as a test.
I would tend to think that the sentence you give should be rendered "the accredited status of the testing laboratory", but I'm making assumptions in order to say that. I'm assuming that the laboratory is a full production facility, doing a job that needs doing using established processed. "Testing laboratory" is the phrasing that describes that unambiguously. I am also assuming that the laboratory provides, for example, some sort of testing that is required for some product or similar, and for the product to be certified the lab must be accredited. It could also be a forensics lab and for its reports to be admissible in court it has to be accredited, or various other possibilities that amount to the same picture of a regulatory environment that requires the lab to be accredited to do its job. However, "accreditation status" would still be an okay substitute, as while it does mean something different in general, the meaning in this case would be equivalent.