First of all - I am sorry for your family's loss and the doubt and anger you are going thro
I dont think as yet you need a private post mortem
you need to speak to someone who would carry one out if it turned out that one were needed
and whether it should be done
https://www.forensicequity.com/forensic-pathology
seven people I dont know - so you can take your pick
I think they will be able to tell you what you will find out from a private examination and even whether it is worth it
or contact Dr Peter Acland who was the west mids home office pathologist who went private whom I did know - I think he is still in practice - even if he isnt it would worth a contact. you would need to google him
or the rcpath may have a list of private pathologists ( doubtful )
almost no-one will recommend one ( because their employers will chop their hands off ) because there is a fear that if things get screwed up the dissatisfied party will try to sue the recommender ( moot and unlikely).
I dont need more detail - but start at the place where he died and ask to see the consultant
You could also book an interview with the coroner ( you will see a bag carrier but it is still useful )
The coroner whose duties are who, where, what from and when MUST feel the answers are obvious. - as those are the legal questions he has to be sure about ....
A post mortem may NOT be the answer to 'now how did this all begin?' -
You dont need to tell us anything else
you need to contact and discuss someone in whom you have confidence.
discuss the possibility of a virtual post mortem ( you MRI the body after death )