I went through this with my mums funeral service several months back.
My father had paid into a plan some years before and had around 8k available . From memory and its not a good time to really have clear recollections , I think the fund did pay out promptly , we went with a director who arranged everything and took care of all the paperwork side of things , there is a fair bit going on in regard and to be honest money isn't the thing on your mind at the time .
I would suggest though having around 10 k aside , with cremation costs ,transport and storage ,directors fees , chapel fees and light refreshments , no flowers , newspapers ,a really basic service, my dad still got an $1800 invoice on top of his 8 K, there was no pressure on us in regard to pay up right away .
you stay well Alex