Hi,
your upper range in searchspace is around 116 bit.

As you don't know in which bitrange the key is exactly to be found it will increase the searchtime.
I don't have the math but I think when using pollard kangaroo and you don't know the exact bitrange (somewhere between 1-116 bit), you will increase searchtime to the same (if not even more) as a regular 120bit key. BSGS will need too much memory and prepocessing time.

You might expose your calculations for the target key so people can audit and you could get more feedback.