It seems pretty clear to me; you are not liable for any 'unreasonable' charges in relation to your training costs. What is reasonable, and what is unreasonable, is obviously a matter of opinion and, in the last resort will be decided by a court, if you cannot come to a compromise with your training provider.
If you have been told that approach and landing charges are in addition to the hourly rate, then these must be reasonable in relation to landing fees and approach fees in general, or you must be advised in advance. If you have not been so told, then you will be liable if it is normal practice within the training industry to add approach and landing fees to the hourly rate for training.
As far as the charges in relation to the flight concerned, they are the initial responsibility of the operator who must then charge them on as appropriate. It seems strange that you have received a bill direct. It sounds to me as if the initial bill went to the operator who then asked the airport to reinvoice you! Query this with the airport concerned. I believe you have no personal liability to the airport for the charges.
If it comes to a fight over this with your training provider, the common law principal of 'quantum meruit' will prevail. A court will make you pay 'a fair reward for services rendered' by your training provider. No court would deem it reasonable for you to have to pay 300 pounds per landing during your training. Bear in mind this is a question of what it is 'reasonable for you to pay your training provider'; it is eminently reasonable for the training provider to pay the airport - that is the charge and they should have known about it. You will end up having to pay a 'quantum meruit' based on approach and landing charges that are normal in training.
This is my tuppence halfpenny worth! Good luck!