In the last ten years the trend has been for airlines to get away from doing medical exams of pilot candidates. In the past they often had medical standards which were more stringent than the FAA, but today for the more part those who still do medical exams are just looking to verify that you actually meet the FAA standards (may be a few exceptions)..

I think it would be hard to find a light cargo / fractional / 135 operator that does psych screening.

I do not know of any regionals that do it.

Several majors still do pysch screening, including DAL and UAL (maybe AA when they start hiring ?). SWA does not.

There is a very high failure rate for those that do it...many folks think that pilots fail because they try to "game" the test to produce the results which they think the airline wants to see. As you may know, the tests are rigged to detect those who try to manipulate them. You have to pass on two counts: 1) Matching the desired profile 2) Honesty when answering the questions.