Why can't this molecule avoid being antiaromatic by becoming nonplanar?
The central atom is approximated to be tetrahedral and thus approximately $\mathrm{sp^3}$ hybridised. The three atoms of the outer ring it is bonded to are approximately $\mathrm{sp^2}$ hybridised as they are all part of double bonds.

If you build this, you will notice that while these three atoms are necessarily in a plane they have different orientations. It is simply not possible to make a planar ring without some significant distortions of the geometry. I thus challenge your professor’s assumption.

I would expect this system to have strongly localised double bonds, i.e. significant bond length alteration, in such a way that the double bond are twisted with respect to each other to minimise strain.