It feels like they put this into contracts to keep you locked into
Notice period locks both parties in, which means that in most circumstances they cannot just outright fire you, but instead either have to keep you for the 3 months, or offer to pay you equivalent lump sum instead. By the same token this also protects the company from having employees leave on a short notice which can be destructive to operations.
But as (almost) every contract in life, if both parties agree it can be modified. Given that they've placed you on furlough and are planning layoffs, I don't see that there would be a big issue with them agreeing to reduce, or even completely waive off the notice period. Talk to your boss, explain the situation and see if you can agree on some middle ground, though honestly, as you are already furloughed then you leaving early should have no impact on the business, and you are only negotiation when you will leave, not whether you will.
Although I would shelve that matter until you have a job offer from another company and are negotiating around the start date. Otherwise you would be taking risks that I don't think are necessary or really give you any sort of benefit.
Can I just leave early and only give my organization 1 month notice?
If I do, what are the potential repercussions?
You can, and there is likely to be very little in terms of monetary/legal repercussions, it's just not worth to chase it across civil courts. But it will be quite highly unprofessional thing to do (maybe less due to furlough and announced layoffs, though some people may not care about the reasons at all), and people who will see it happen in the current company will likely remember that in the future if your work paths ever cross again. So it is always best to try negotiations first, before reaching for the nuclear option.