I suspect that the white core wouldn’t be white anymore, if you know what I mean. Cutting the mat board should definitely be an option.
As far as the felt goes - one of the members here tested it on another laser. The Glowforge will be able to do the same - her results are here:
https://community.glowforge.com/t/laser-cut-100-wool/4447?u=jules&source_topic_id=6432