Does anyone know if this new reprint of the Core Set will contain any changes? Like more copies of each card, such as been done in other expansions?

It is highly unlikely.

My guess? There is absolutely zero chance the Core set will contain more copies of each card. If it had, I believe FFG would've made an announcement ahead of time like they did when revising Secrets of Arkham (as well as raise the price). Core sets for all of the LCGs have been reprinted before and none have been changed in the past. I don't see it happening.

They recently changed Secrets of Arkham because it was the last deluxe expansion that only came with 2 copies of each card (due to it being quite old comparatively speaking). It now has the same format as all the other LCG boxed sets.

Is there a way to get more copies of the Core Set cards, short of shelling out for more copies of the Core Set box (or trading)?

Alas, no.