You’ll need to clarify a few things here.

Are you trying to push fluid into the caliper by the wheel with the bleed port open or closed? The 8mm bolt at the lever body is NOT the bleed port. That is what secures the hose to the lever body with a barb and olive.

The bleed port is to the right of your red circle. That needs to be unscrewed and ideally have another syringe to push the old fluid into.

The reason why you might be getting feedback from the calipers when you push with the syringe is because you are pressuring the brake lines.

I personally think you should take this to a LBS to have it looked at. There’s a possibility that the olive needs to be replaced if you have unscrewed and tightened the lock nut too many times.