Would be interested to know how the authorities knew what height it should be at on a climb out. Are they suggesting that it was performance-limited in some way?
What you do is check the obstacle limitation surfaces against where the crane is. If it is below the surfaces then the aircraft should have cleared it by whatever the standard minimum climb gradient acheives at that point. Note - most aircraft climb far quicker than that.
If the crane is higher than the obstacle limitation surfaces then it should have been NOTAMed. However this aside, it is very easy to get the aircraft flight manual, calculate the climb gradient and reduce it by a factor (net performance) and one has the height at which the aircraft should have been at that position.
There are minimum rates of climb for every category of aircraft. Even permit to fly aircraft have to demonstrate a minimum rate of climb.
Pilots are often unaware of or ignore the standard obstacle limitation surfaces - see other topic regarding landing distances - and as I said some time ago in that topic - when something happens, your ass is on the line if the authorities (or the insurance company) can show that you did not keep the required separation from those obstacle surfaces!
Regards,
DFC