There are two problems here, and only one of them is about money.
The first is evidential. If you cannot show where the four hours went, you have no basis for the conversation with the client and you will lose it. Record time against the file, and separately record time spent on research that benefits the client but attaches to no single entry. The second category is the one that habitually goes unbilled, and it is often the larger of the two.
The second problem is contractual. A fee schedule written before the work changed does not bind you to lose money indefinitely; it is simply the schedule you have not yet renegotiated. Take the record you have built, show it, and ask to be paid hourly for classification specifically. If they refuse, you will at least know what that relationship costs you.
