Is the polarisation of halogens of higher atomic number easier because the electrons are further away from its core?
Short answer: yes.
The Wikipedia article on Polarizability has a nice section on this, which I would like to supplement with further quotations from its cited source.$^1$
pg 24

Since electrons are charged particles they respond to electric fields.
...
The electron cloud is said to become polarized in response to the electric field. The ability of the electron cloud to distort in response to an external field is known as its polarizability.

pg 25

We define the polarizability of a molecule as the magnitude of the dipole induced by one unit of field gradient, which works out to be in units of volume. Often, the larger the volume occupied by the electrons, the more polarizable those electrons.

My understanding is that it's less that the electrons are farther away from the nucleus in higher elements, but rather that they occupy a larger volume and are therefore more greatly affected by the electric field.

There is probably a nice analytical expression for the relationship of the polarizability to the terms we're used to seeing in the Schrodinger equation, but I'm not aware of it. In my experience, it is typically calculated numerically using perturbation theory. I did find an example where they worked through the perturbation solution for a particle in a box being subjected to an electric field here, section 4.1.1.

$^1$ Anslyn, Eric V. and Dougherty, Dennis A. Modern Physical Organic Chemistry. University Science Books, California, 2006.