It was Biden/Harris. The officer was turned into a martyr and held up as proof of the brutality of the riots. You can find some examples of such by searching for 'officer killed in capitol riots' on YouTube. For a company that claims to care about misinformation, they're perfectly happy having hundreds of videos up both stating, defacto, that he was killed - and continuing to politicize it.

And while I have become extremely skeptical of the competence of our government and agencies, I can safely assure you they are more than competent enough to at least determine whether somebody was murdered at the Capitol on a given day, or not.

It explicitly said in the wikipedia article "The following weekend, Trump ordered flags to be flown at half-staff at all federal buildings, grounds, and vessels for three days". Biden didn't assume office until the 20th.

Flags being flown at half-staff is a pretty normal event, not a rare honor. The ceremony was held on February 2nd. The picture chosen for the Wiki article entry is from that ceremony.

Perhaps, but there seems little reason to suppose had Trump/Pence still been in office the decision would have been any different. Just seems you're reading more into the story than is justified.