You don't need to demonstrate anything other than simply demand the content be restored. You do risk getting sued by the folks who issued the original take-down notice, but you already know that your are in the cross hairs of whatever company issued the take-down request in the first place.
The person you need to demonstrate that you actually have a legitimate claim on this content is simply yourself, so far as being bold to demand that it be restored with a counter notice. If you get such a notice, take a moment to pause and really consider if it is legitimate or not. If you think it is (perhaps even consulting a buddy who is a lawyer if you really want to be sure), don't hesitate and demand it be restored.
I am also suggesting very strongly that most often you won't even get anything else happen after it is restored, because most of the trolls don't want to spend money (it does cost money to file a lawsuit) unless they know they are going to get something from it. By filing a counter notice, you already are showing a little bit of backbone, so they are further hesitant to just automatically go onto the next step without doing a pretty good review of whatever you demanded get restored.
I'll also note that if it goes to the next step of an actual lawsuit, "actual damages" are a whole lot easier to assess so far as legal fees are concerned, which can be a part of the judgement. It doesn't take a separate action here BTW, unless you really intend to file a lawsuit over the initial takedown request where they refuse to take any further judicial action. Note here there is also a statue of limitations that limits how long the company can wait before filing the lawsuit after you demand the content is restored. Repeated take-down requests also show judicial abuse and can get some fines and statutory damages.
Don't get me wrong, the DMCA is a horribly written law, but in this situation with the take-down notice process, it really is one of the better parts of this particular law (not perfect, but better) and an improvement over the earlier process of fighting lawyers through snail-mail responses and cease & desist letters. The take-down process also doesn't stop a lawyer from sending a formal cease & desist letter after you demand that the content is restored too, so there are additional steps that the people like Qualcomm, BMI, or Disney could take before even filing a lawsuit.
The alternative, what you are refusing to acknowledge, is without this first step and goes straight to the lawsuit. At least with the DMCA take-down notice there is a small speed bump along the way that can give everybody (both the target of the request as well as the person making the request) a chance to review what is going on and potentially flag abuse. If you really think something better could be done, please express that alternative other than going straight to the lawsuit phase. I also disagree that this puts the burden hugely on the defending party.