Only the first murder could concievably be second degree or felony murder as in striking out when startled, but as soon as the first fatal blows were landed all that followed became a premeditated choice.It is incumbent on the Prosecutor, imo, to go for first degree in this case, as there are suggestions it could be. The defense can try to claim that there was no intent, but to make the jury believe them, they will have counter the active arguments put forward by witnesses for the State.
I've seen it theorized here that he might have still been practicing his criminal doings, or that he came with an idea of something other than murder. If the Jury believes that Kohberger did indeed have a knife with him that Sunday, they aren't going to buy that he had zero intent. I don't know enough about ID law to know whether felony burglary would automatically instruct the jury on this point: if there's a felony murder rule (and I think there is, from discussions here early on), then it's first degree murder. I think that's why there's that fifth charge.
State needs to prove that BK came inside the house and then, with or without intent, if they prove he killed four people, it's going to be in the first degree (I think; imo; IANAL).
OTOH, the State may also be able to show further signs of intent. And I believe that's where the big pile of evidence is going to be heading. Juries are persuaded by all manner of things (individually and as a group), so it behooves the State to provide evidence of more than one. Prior encounters with any of the victims; attempts to "stalk" or "get to know" or "interact online" with a victim (even worse if more than one victim), etc. Prior trips to the house (seen at parties). Pretty long list of what they must be investigating.
I imagine widening circles, ripple effect.