Tbh, I don’t think you really understand how the non-rhotic accent works. In this case, the /r/ would be fully pronounced, as it would be at the start of a word. Say bread, elongate the r and skip the ed part and you have what it sounds like.

If you’re very used to hearing the bunched r, the British version still might sound softer, but even in the USA (where most people use bunched r) it’s still common to hear an r made with the tip of the tongue behind the teeth (upper or lower).

I’m ignoring the other r sounds, but you do find a lot of them across the various regional English accents.