First time I’ve cut my hair like this since highschool. I missed it!
It depends on your hair, some people just have hair that is terribly resilient. My hair is naturally black and I had to bleach it so many times so get it to lighten. But then my sister also has very dark hair (though hers is more dark brown, whereas mine is black) and her hair lightens very easily.
Sometimes you can get those boost dyes made especially for dark hair, a friend of mine with naturally dark hair had great success with the burgundy one of those (I’m pretty sure it’s a Schwarzkopf brand one, if you want to try dying it yourself).
However, if you’re going to a hairdresser, and they’ve already tried without bleaching it, then maybe bleaching is the best option. I’ve been cutting, dying and bleaching my own hair for years and never had a disaster, but in saying that, I’m not certified, so always trust your hairdresser to know best as they’ve been trained.
An alternative is perhaps bleach just foils into your hair, instead of bleaching all of it, and put the burgundy over all your hair, then you won’t damage your hair as badly and you’ll have the burgundy highlights among the black/burgundy hair.
Hope that helps <3