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

I would rather have a higher sidelobe amplitude, if I can also not have deep
notches in it, than lower sidelobes with deep notches, unless of course the
sidelobes can be made to be insignificant.
And at this point, I'm not sure how low they would have to be, to be
insignificant.

A small dish is possible, but given the wavelength, "small" is not really
all that small.
I don't get a choice on wavelength.
In order of importance, tight beamwidth, low sidelobes, "smooth" sidelobes,
and gain.

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.

It probably would, but I'm just at the back of napkin stage on this project.
I won't know what I need really, till I do some testing.