This actually seems like an important choice so I'm going to try and think it through. Doing this is the first step toward claiming protagonist status- if we let Ichigo do it, we're taking a decently important character beat and throwing it off onto another character to take care of. In addition, I can see Tatsuki, as Orihime's friend, definitely thinking that if anybody is going to put Orihime's brother's soul to rest, it should be her. There's definitely value there from a character standpoint. There are also potential experimentation bonuses and gains involved in doing it now, as soon as possible. It could potentially also be seen as Tatsuki passing the buck to Ichigo, which Orihime wouldn't care for.
On the other hand, we're already traumatized the fuck out due to the reveal of the nature of the Hollows to begin with, and I really doubt having to go through the process of purifying one without sufficient prep time. Ichigo definitely does have some rights to take care of this himself, to an extent. And I think that the personality argument is decently sound; there is a really good chance that while this isn't the definite marker of whether we become somebody more like Orihime- compassionate, cares about others, all that good stuff, or a HARD GIRL MAKING HARD DECISIONS and I definitely want to go down the former road if at all possible. And to add to the emotional component, Tatsuki really cared about Orihime's brother, and I can definitely see this being too much for her. There is also minor risks involved.
Overall, I tend to look for characterization arguments, and I think the characterization arguments against are much stronger than "for". (The one about Tatsuki being Orihime's friend has some merit, but is kind of overwhelmed by all those reasons against, and I somehow think that if Tatsuki explains her perspective to Orihime, she'll be considerate enough to not consider it "passing the buck" at all.)
[x] Ichigo should do it. He knows more than any of you why Orihime's brother can't remain as he is, why he can't continue to be a threat to his sister.
We're not being a coward. We're being a realistic version of Tatsuki that could be setting herself on the right route to the future; we don't need to be front stage if being front stage doesn't quite make sense, takes way some mild character beats, and could potentially be opposite the way we want to go.