I'll tell you how to convince all skeptics - or force them to admit the reality of LENR.
First, develop a high powered LENR system utilizing the know how that's already floating around the community.
Secondly, test the heck out of it - again and again. Once you know it's repeatable and that you can trigger the reactions pretty much at will, then start changing individual parameters to learn what's important and what's not. Importantly, perform experiments testing the SAME materials, components, and fuels from DIFFERENT sources (the more commonly available the better) to make sure they will still allow for the same POWERFUL effect to be reproduced.
Thirdly, start working on how to make your system simpler, cheaper, and faster to build.
Fourthly, design an experiment that will show the massive excess heat in an obvious way. A few things to do: use multiple control reactors, use various instruments to measure temperature, flow rate, or power consumption. Think about how skeptics will attack every little detail and make it impossible for them to criticize your measurements. For example, if they think that your thermocouples are wrong use analog thermometers as well in addition to other techniques to measure temperature. When it comes to flow rate, don't simply use a flow meter in the circuit but utilize a basin with a known quantity of water for the input and measure the output in another basin. From the start, assume they are going to claim that you've screwed everything up.
Fifth, prepare absolutely detailed plans with every last bit of your knowledge that could help replicators to be distributed when your demonstration takes place. Make sure you list every possible parameter and make clear warnings about the areas in which varying from the plan could result in failure. Treat potential replicators like brain dead idiots and explain everything step by step.
The problem is that almost no one wants to do the above.