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Translating or color-remapping isn't going to let ZDoom magically have palette colors that are not in the palette. You want your sprite to have these shades of mauve, either you keep them as PNG and tell people to use GZDoom only, or you create a custom palette with the colors you need in it. The Doom palette isn't the color you need, simple as that.
As Gez said, you will only get proper colors in GZDoom, Zdoom will always butcher the colors till it matches Doom's original palette. There's no way to get around that in Zdoom, as it uses the software render.
So with that in mind, with what you want to achieve, just use GZDoom. You also don't need to convert pngs for them to work in GZDoom, btw.
Then do not convert to paletted sprites using the Doom palette? Really, it's the entire issue here. You're converting your horse sprite to the Doom palette specifically. Why are you doing that?
The renderer itself is the software renderer. You can tell because the lighting/pretty much everything sans the HUD is 8bit. And looking most peculiarly like Doom's column renderer, not a 3D hardware renderer.
I ask this because you don't exactly seem to be on the same page as what everyone else is talking about, which is odd because you seem to have been changing settings you wouldn't have known about otherwise. GZDoom defaults to its own specific hardware renderer, and has no menu setting for the [wiki=CVARs:Display#vid_renderer]vid_renderer[/wiki] mode.