Again, to be clear, this anecdote is not verified. It is extremely poorly sourced, not much more than an urban legend. The only attribution for the story dating to the 19th century is a claim that someone heard it from the sultan's physician's son, and that claim is put forward more than 40 years after the events.
The existence of donations, verified or otherwise, does not independently show that the British government attempted to prevent aid from reaching Ireland because they were embarrassed about how little they had contributed. That is the claim that OP puts forward with their story, that is the claim that I'm responding to. I'm not questioning that others did send donations or that some of those donations exceeded those put forward by the British government.
All of that is true, but it being true does not justify perpetuating unsubstantiated stories that happen to support the same conclusion. As you have amply demonstrated, there's enough good evidence in favor of the conclusion that we don't need to rely on bad evidence.