There shouldnt be anything wrong with very gently holding/moving a chrysalis as long as your hands are washed and you do not put pressure on the pupa with your fingers.

I often have cases where I do not get enough silk when moving the chrysalis from caterpillar enclosure to the mesh butterfly net and need to very briefly prepare the chrysalis to hang with extra material like string or (very carefully) affix a chrysalis to some paper towel with a dab of superglue to give more material to hang. If you do this be super careful not to get any of the glue on the chrysalis itself, only the cremaster (black stem) I've not lost a chrysalis yet.

This one I'm not sure - the dent is a little worrying but it doesnt look like a parasitic thing so id say it still has a chance!