OP, I think you're confused about what the various documents mean and what you need to prioritise.
There is no "next of kin" named on the death certificate, whether interim or final. It just names the informant... your aunt as she registered the death. The death certificate gives you no legal standing.
Please don't use the phrase "next of kin" when dealing with organisations or your aunt... it has no legal meaning. Refer to yourself as the Administrator of your father's estate.
The Coroner's role is merely to determine the cause of death. They won't get involved in anything to do with the estate or probate. If there's an inquest you'll be invited to attend as you are an Interested Person.
As your dad died without a Will, the intestacy rules apply. Look them up. In the absence of a surviving spouse or civil partner, the estate passes to any surviving children i.e. you and any other children your dad fathered.
You absolutely need to apply for probate (a Grant of Letters of Administration) as soon as possible. That is the document which gives you the legal authority to take ownership of his assets, bank accounts, etc. You need to get this document before your aunt does.
You only need to estimate the value of the estate to apply for probate . If your dad's estate is likely to be under £325,000, there will be no inheritance tax to pay and the process of applying for a grant will be quite simple. You can do it online.
Finally, your aunt could potentially make a claim against the estate but will only succeed if she can show that she has a financial dependency upon your dad, that he was giving her money or supporting her in some way. Nevertheless her claim could tie you both up in legal costs so, if this happens, you might want to negotiate a sum to pay her off.