There could be room for something like the International Red Cross, supported by downtime and uptime services.
I fully agree, but it absolutely should not be mixed with the situation with the Southern states.
Instead it should be a proposal made to the international community on its own merits, and possibly based on Switzerland for the same reasons as originally - neutrality. For it to work, the Red Cross must be seen as strictly neutral, and even though the battle of Solferino might not come to pass any more the reasons for such an organization still remain valid.
Cali could donate the funds to set up the organization, make available the original charter and all subsequent modifications up to the transition, it could teach how to produce whatever medicines and vaccines are possible with the current technology, etc., but it shouldn't try to take a leading role in it - or even participating as more than one of the founding signatories - to avoid impugning the organization's neutrality, and this stance should be publicly known.
If there is one modification that the Californians might try to implement is that the symbol isn't associated with a current religion, avoiding much drama later on (and incidentally satisfying their own atheists and non-Abrahamics, but that might not be something to openly tell the other nations). Maybe the
Rod of Asclepius, and call it the Red Rod, this could kill another bird with the same stone making certain that people wouldn't mistake the healer's mark with the
symbol for the Greek god of merchants, gamblers, liars, and thieves (also of messengers and shepherds, but you get my point).