I purchased my replacement core from Winnebago and had a local RV service tech replace it. It does look impossible to remove, but a flexible, slender person can get it done without removing the front cap or disassembling the entire firewall. Cost me about $300, 3 years ago. The dealer wanted twice that.
IMHO, the leaking is caused by dissimiliar metal corrosion and poor manufacturing. Don't know where your leak is, but mine happened right at the end of the core where the connections were. A lot of people have had this core fail in Ultimates, Journeys and Horizons....and those are just the ones I know about.
You can bypass the core by connecting the two coolant hoses together. I carry a plastic connector just for that occurence. It won't provide heat obviously but it will give you time to repair/replace the core and still drive the coach.
As for the charcoal in your water, the other posters nailed it.
Your coolant may be wrong or may be old, but in the end it's unlikely to cause leaks all by itself. As I said, I think it's just too many different pieces of metal coming in contact with a really good conductor.
Good luck with your repair.
2017 Cedar Creek 38EL/2015 Silverado 3500HD