My husband very nearly died of sepsis a few years ago. I'd just had a baby and had a small child. We had a strong case for medical negligence as, when he was in full septic shock, his blood pressure still wasn't taken. His skin was discoloured and he could barely walk. He collapsed a few hours later.
Because of the very fast progressing nature of Sepsis, I feel that the fact that he was seen and discharged 2 days prior probably wouldn't amount to medical negligence because he could have been reasonably well, or at least not fatally unwell, at that point. However, with the sheer amount of info around about Sepsis today, I would expect all medical staff to not only be vigilant, but to warn patients of what to look out for and to seek help if symptoms worsen. As sepsis patients can also suffer from confusion, this should ideally be written down and communicated to relatives if possible.
We chose to ask for a review in staff training and processes and they agreed to that. We decided not to sue because my husband survived and the NHS ultimately saved him and became he made a somewhat good, albeit not complete recovery. Also we didn't want the stress nor to take more money out of the NHS as we believed that it was the cuts in the first place and stretched staff that caused it.
Our situation is obviously different to yours. You lost your father and a husband. My gut feeling on this is that you've not quite met the threshold for medical negligence (it's a high threshold), especially with the fast changing nature of Sepsis and your father's pre existing health conditions which could have clouded the issue. I have a legal background, although have not worked in the profession for many years. This is just my opinion however.
In making your decision, you could always read the Nice Guidelines for dealing with Sepsis and see whether they followed them. You could also contact The Sepsis Trust for advice. They are a really helpful organisation. I've previously done some voluntary work for them and helped train health professionals on Sepsis.
I am so sorry for your loss.