In cases which go to trial in the Crown Court, the defence have a right to examine physical exhibits before the trial. The reason is that these are just as much evidence as witness statements are.
Coroners are a law unto themselves or, rather, to laws which make their powers and their procedure different from that in criminal courts. There is no 'defence' since nobody is on trial. It is an inquest conducted by the coroner and nobody's representative has the right to ask questions of witnesses, though the coroner may allow them some, and usually does.
So the short answer is no. The private detective has no standing. If he is employed by the family, he still has no standing. It is possible, that on informing the coroner, he or she might allow, at the coroner's invitation, someone other than police officers and police doctors to examine exhibits but only to help the coroner in the inquest into the cause of death. Sam Spade himself would be unlikely to satisfy the coroner in that regard !