The P800 has superior Maximum Black compared with the previous models, and in principle I would think you should be able to achieve excellent tonal control with the P800 using custom profiles, or canned profiles for this printer and of course careful luminance editing. I'm surprised you would be less satisfied with the P800 given the improved dynamic range and the remaining possibility to exercise good control over shadow detail, etc. Of course much depends on the details of what you mean by "proper image rendering".
well, I am very satisfied with my P800, just wanted to say, that it is different than K3 printers, behaves differently, sometimes better, sometimes worse than 3880
I really like MK density, now with simple matt papers we can achieve black densities reserved earlier only for Epson Hot/Cold Press papers on K3.
I like PK density, but not its look - it looks like oily tar when light is reflected where pure PK ink is used with evident border from lighter tones, it spoils some prints on barytas. Although, K3 PK is not that dark, I often prefer it with my 3880 over P800.
MK behaves better on "waxed" fine art, like HM PR Satin or Solution Bright White Satin, it does not smudge under your ones finger as happens with K3 MK even after several days
however, MK behaves differently in max density area. For K3 or K7 (Piezography), curves for L close to 0 area could consist of several ink tones with different proportions (mostly MK, but also LK or even some LLK). It is not recommended with P800 MK HD, since black density goes down when black ink is mixed with some lighter tones. etc etc just different printers.