Without knowing for sure, I can only give you advice based off my personal experience and also extensive research..
But it will be different for everyone. It's not necessarily the labral TEAR that is hurting you. I ran for months on a labral tear and had little to no symptoms. The impingement is what started to hurt me. It's complicated to explain but I'll try to simplify it. You glute pain and hip pain and groin pain are all secondary/radicular to the labral tear and possible bone impingement. The impingement is clinically known as Femoral Acetabular Impingement (FAI, if you want to Google it). You can have two different types (or both, I have both). CAM impingement or Pincer impingement. Basically both involve abnormal bonegrowth and/or adhesions on either the femur, the hip bone, or both bones. When you hip flex (during running, sitting, walking, etc) your bones hit each other and it produces abnormal rotation inside of the hip socket - hence "impingement." This impingement causes bad contact in the hip capsule and surrounding it which tears away at your labrum. That is how MOST tears occur.
SO, if you do have a form of impingement, which is highly likely, anything involving a decent amount of hip flexion is going to produce symptoms or radicular pain (groin, glute, IT band, etc). What happens is your pelvic muscles become dysfunctional; because you cannot feel the extent of damage that is inside of the hip capsule, muscle such as your hamstring, glute, quads, TFL, etc become chronically tight. Muscles that seem to be the worst for this condition are hip flexors, glute, and psoas and groin.
Bike does not work well. A lot of hip flexion and then add in resistance.
Elliptical is ok for some and not for others.
Running = no
Walking = ok for some not for others
Lower Body lifting = BIG NO, this will cause more damage, esp. a squatting position/motion
Swimming is usually ok, pushing off the wall may or may not agrivate it. I was usually ok. Pool running was ok until about 2 months then I was forced to stop.
In the end, just listen to your body. Bike might not work for me but for someone else it might be ok. It depends where the impingement is and during what part of hip flexion/extension it bothers you. If it causes pain or tightness, stop. If you feel ok and the pain does not increase during or AFTER exercise, then you are ok.
Labral tears are an active recovery, but minimum of 3 months post-surgery before you run again. I just had an operation. I am on day 19 of recovery. Much less pain. Great ROM. But I'm not running.
I asked what state you are in because If you are near NY I would suggest a few Dr's to see. I REALLY would recommend doing research before spending money on imaging. If you are having just any old ortho guy look at you or prescribe imaging, you may not get treated thoroughly, I'm just warning.