simply weighting it could at the very least be able to demonstrate that something very unusual is going on, without excess heat or nuclear products.
Well, weighing it is not simple.
In an electrochemical system you can measure the amount of hydrogen with the orphaned oxygen method. That gives the weight, but not the concentration, because the distribution is probably not uniform. You cannot weigh it directly. By the time you transfer the sample to a micro-scale, much of the hydrogen would be gone.
Although the density may be great, most of the hydrogen is probably in a narrow band close to the surface, so the total mass is small.
Unless you detect excess heat, you can bet nothing is going on. As I said, there is no point to looking for nuclear products in that situation. You are wasting money and time. It is possible some reactions are so small they do not produce detectable levels of heat, but you cannot tell the difference between that and a complete failure.