There are a number of standard tests that are run for lupus and 1 additional one for MCTD.
MCTD is confirmed by an anti RNP test.
Here is the LFAI site with a full description of the tests normally done for lupus and MCTD.
Sed rate and CRP are tests for general, systemic inflammation and should also be done. If you have any muscle involvement a CPK test should be done as well. None of these are tests specific for any given disease but help to define what the problem is.
Any experienced rheumatologist should know about these but make sure the doctor is experienced in treating lupus and other AI diseases like MCTD and myositis. Depending on your presentation and other test results Ultrasound, mri, emg, or muscle biopsies could be ordered. You will be an expert at the end of all of this.