Question on bleaching dyed black hair?

I dyed my hair a solid jet black about 2 months ago. I have naturally dark hair, but the black just made it more striking.

Anyway it has faded out mostly as I can see brown undertones coming into it again, but it is still mainly black.
What I want to do is lighten my hair a shade or two, not much, and then dye it dark brown.

I don’t want to use hydrogen peroxide (I was told that this bleaches black hair really light blonde very easily), but I would rather use box dye kits, as I feel they are less damaging and have had experience with them.

I live in the UK (London), so can anyone tell me their experience of the best brands of blonde hair dye for bleaching black hair, something available at Superdrug or any other major chain store? I just want to take it literally a shade lighter, just so I can dye it dark brown.
I have used Live Colour XXL in other colours before, but not the platinum blonde, is this any good? Any suggestions thanks.