Frequency is 2.4 ghz, plane is made from radials about 4x wavelength
(can't change this), and I can place a monopole antenna anywhere up to
about 5 wavelengths away (up) from the center.

I would like to maximize reception[1] in the angles close to the ground
plane (~30 degrees) and/or minimize the null at 90.

Can I shape the pattern by appropriate placement of the antenna?

[1] yes reception; this is for a radio application pretty far from normal
amateur radio (rc control of a flying vehicle - so the antenna will
actually be mounted *below* the ground plane), but

a) there will also be telemetry in the opposite direction using the same
antenna; and

b) the rc guys don't know **** about antennas, it's all "I bought this
one and it worked great"