For the vibrant purple ends, you’ll need to remove the existing colour then lift it to a pale yellow, 9/10 before toning to neutralise, then adding your target shade. If you try adding purple on anything darker, it won’t look very bright.
If you use a direct dye on freshly bleached hair, it might grab onto the cuticle layer, but it’s equally likely to fade quickly, if it’s quite porous and could lead to patchiness. I find it takes 2-3 applications and then it will last much longer between re-applications. The client will need to use a sulphate free shampoo and wash once, not twice, in luke warm water to maximise longevity.
I’d use a direct dye on the ends as they’re much generally brighter than permanent colours. Guy Tang’s Mydentity colours look good and you can buy them at Salon Services.