You are correct, the car will lose weight as you drive, and in real terms the ratio between the weights of the car and trailer will change, but that is not how the industries conventional towing ratio is calculated, it does not use measured weights, it uses the trailers MTPLM divided by the cars Kerbweight. Both of these figures are paper figures, and in theory represent the worst possible scenario for your outfit.
In practice and provided the caravan is not overloaded, the outfits real world weight ratio will always be better than the industry calculation even when the fuel tank empties, because cars are rarely driven in their kerbweight condition when towing they invariably have other stuff or passengers or luggage that ensures the car will be heavier than its paper kerbweight.
This amply demonstrates one of the inherent weaknesses of the industries calculation, it does not take into account real world variables.
Even though the car loses weight, the overall balance of the car would not be affected very much, at a difference of about 60kg, it will be virtually the same as the effect of losing 1 adult passenger from the back seat. You would have to be an exceptional driver to be able to notice such a weight change when towing.
If as in the scenario you did provide, where you started the journey and all seemed well, but you found the outfit less stable at the end of your journey when the fuel load had diminished, no well matched outfit should be that sensitive, and it points to a poor match to begin with.
I'd suggest it would be more likely driver fatigue or present speed, other road conditions than the loss of fuel weight.