I don't know of any formulae that do/give this sort of thing. It would have to be a specialist text book of some sort. The simplest answer is to run it up on HFSS with parametric sweeping of the slot positions. I am surethere are other equally capable full field simulators as well. The trouble is these are all big expensive software packages, but that is just the modern world we live in.

You need big bucks to play in the big league!

I'd expect this is very high power since these spec's warrant a low cost printed circuit patch array considering the 4% bandwidth. If it's not high power, then whoever said to make a slotted waveguide array instead of a patch array is wacky.

There are computer codes for designing this.
Ansoft HFSS is one of the best. We've used it on waveguide arrays.

You need to add sidelobe spec's and polarizations to set up the problem, plus a physical footprint.

kch