BA only offers one flight a day to Catania; I checked.
We did that Heathrow to Gatwick transfer to reach Venice from Seattle on our honeymoon 22 years ago. We were flying business class on miles, our first time flying abroad together, and my first time on British Airways. We used the National Express bus for the transfer and it went just fine. According to the Heathrow website, the journey time for that bus is 70 minutes. But if you would be more comfortable with a taxi, do that. As long as your incoming flight is reasonable on time, you should be fine. The good news is that security at Gatwick is usually easier and faster than security at Heathrow.
We have stayed with British Airways and flown to Europe with them at least 12 times since then. There have been a few minor glitches since the pandemic, but all airlines are struggling a bit right now. I have read on FlyerTalk that BA is juggling the Gatwick schedules, particularly with the long-haul flights.
https://www.flyertalk.com/forum/british-airways-executive-club/2153667-ba-mainline-ba-euroflyer-gatwick-long-haul-short-haul-winter-2024-2025-a-5.html
I have had no trouble reaching BA Customer Service on the phone the past 2 years (in 2021 it was difficult, with hold times up to 60 minutes, when they would cut you off. That does not happen any more).
The trick is working through the phone menu so you get an actual booking agent. They have always been professional and patient, providing exactly what I asked—-or in one case, making suggestions to a better solution to my request. There should be no need to demand to speak to a supervisor. They truly want to help you, so give them a chance to do their job.
But with only one flight a day into Catania, and that one from Gatwick, there is only so much they can do if you do not care to chance the airport transfer. One option would be to fly into Palermo instead—-that flight departs from Heathrow. But there may be only one flight a day, and it departs in the morning, so you would still need an overnight at Heathrow (not a bad option if that works out for you; the Sofitel in T5 is quite nice). Or you could fly to Naples and make your way to Sicily by ferry, but that is time-consuming.
The other option is a different long-haul flight, if there is one that flies into Gatwick. Looks like they may still have flights from New York. But that would require an extra plane change, and I doubt you want that.
You are entitled to a refund if this schedule change is simply unacceptable. But check and see what your options are with other airlines, and the cost, before you request that.
Good luck, and I hope you arrive at an acceptable solution.