Bombed an academic interview, should I do anything about it?

First of all, you should cool down before you take any action. Clearly, the interview was an emotionally intense event for you, so don't do anything in the spur of the moment (and "a few days later" may still qualify as "in the spur of the moment" in this context). Let a few more days pass, and see how you feel later.

Second, seek the guidance of a trusted person or mentor who is aware of the details. Maybe your advisor can feel out to what extent you have really bombed the interview quite as badly as you perceive it? Note that especially for interviews on grad student level, the committee will be positively used to people giving bad presentations and not being quite able to argue their case. Unless you have made a really monumental blunder, the committee has probably already forgotten about the details of your interview by the time the next coffee break rolled around.

Which brings me to the third point: no matter what you do, don't write an email reminding people that you did poorly or excusing yourself for not doing better. Don't try to argue your points, either. This won't make your interview better, but now people may also think of you as somewhat unprofessional. If you feel you need to write an email, keep it professional, thank them for the opportunity, and ask for feedback.

Any alternative action that you would recommend? Or should I just let it go?

Letting it go is generally the right course of action in these situations. I have personally done my fair share of bad presentations, bad interviews, and (in retrospect) crappy paper submissions. Learn from them and improve for the future. You are a student. Mistakes are to be expected. Life will go on.

Your question reads:

I came across to them as an impostor who was simply winging things/saying random rubbish.

But I wonder what the committee said that made you feel that way.

I know you feel like your answers were on the mark, but I'm going to side with the committee here and assume that you weren't demonstrating the kind of mastery and expertise they were hoping to see. This doesn't mean you were an "imposter" or that what you said was "rubbish," it simply means I'm assuming they know more about this subject than you do, and they were hoping that their interviewees had a stronger grasp and mastery of the subject matter than you displayed.

I've learned the hard way that it's possible to know just enough about a subject that you can – how does the old expression go? You know just enough "to be dangerous;" that is, you may think you fully understand the concepts, but you don't have all the details ingrained enough to sound like an expert when put under close scrutiny.

I'm guessing that they didn't really regard you as an "imposter," but more like someone who had some more learning to do and wasn't yet ready to receive the grant.

As for what to do next, I would avoid your third solution, which would be to prolong the debate outside of the interview. That would likely rankle their ire even more.

I wouldn't apologize for wasting their time, either. That sounds too defeatist.

Your second solution aligns most closely with what I'd recommend, but I wouldn't ask busy people for specific feedback. Instead, I'd recommend sending one or more of them a thank-you letter for pointing out that you still have a lot of learning to do, and letting them know you intend to keep studying the subject and developing a keener mastery of it.

Perhaps you can tell them it was a humbling but eye-opening experience, and you intend to leverage it to further your career, your goals, and your aspirations.

I would think that the committee members would find it hard to argue with such a humble approach, and some of them might be impressed by your willingness to learn more.

Important interviews can be very emotional business. Also, performing well during such an interview is not an easy skill. I think you are not the only person to learn that "the hard way". Everyone experienced a failed interview, and the committee people have seen dozens people to fail. You are no exception.

Instead of worrying about the bad parts, you need to take this as a precious lesson. An interview is not just about seeing what you can do (although it is also important), but also finding out what you cannot. The interviewers need to see how far your understanding goes (and what happens beyond) and how you act outside of your comfort zone. If the interview is tough and you feel lost, that does not necessarily mean it is going bad—it simply means they have reached your limits. What matters is how do your limits compare to others'.

Certainly, this was not the last important interview of your life/career. Do not regret the failure; embrace it and learn from it. Next time, you shall be ready for this situation (which caught you by surprise this time and got you off balance).

Also, there is no reason you should feel like wasting the time of the professors on the committee. They agreed to be on the grant committee and interview the candidates to pick the best; interviewing you was a necessary part of the process.

Finally, regarding your next step: I would recommend against contacting the committee members. Do not further emphasize this episode. As was said in other answers, they probably forgot by now, and the only potential effect the email might have is to remember you as "the guy who failed". If you get to meet them in the future under different circumstances, you can talk about it perhaps in an informal setting and maybe even have a laugh together over the whole story.