What sort of antenna offers narrow beamwidth, combined with smooth
sidelobes?

I don't know what you mean by 'smooth sidelobes'.

Sidelobes are artifacts of truncation and/or lack of continuity, in the spatial
frequency spectrum of the aperture.

There are lots of ways, in hardware, to mitigate sidelobes. In an array this is
done, conventionally, by amplitude weighting of elements. There are
non-conventional ways of course, that are beyond the scope of this NG.

As elelemnts, there are moderate gain dipoles that have no sidelobes. A
conventional solution, for example, is a Landstorfer element, which is a curved
wire antenna that looks like a Gaussian in outline.

Perhaps a bit more specificity would help.

73,
Chip N1IR