How much does an undergraduate degree matter for an academic career?

I will give some answers from an American point of view (in math), but probably this is widely applicable. The answer in short is that your undergrad degree means essentially nothing at this point. It's only role was to get you where you are now.

Now, for some of your secondary questions.

If there are a large group of equivalent, good, candidates for one position, do funding bodies and interviewing committees look at what's different between everyone?

Yes, but your undergrad degree is weighted almost nothing in these considerations. About the only case where it can help is if someone on the committee knows you from undergrad and has a positive impression from then. The primary consideration is your research (both how good it is and how good of a fit it is) and expert opinions of your work (e.g., recommendation letters). Secondary considerations may be where you've published, how active you are in giving talks/going to conferences, impressions of committee members who know you, previous awards you've gotten, and other professional activities. (I'm assuming the things you're applying to don't involve any teaching.) Once you've gotten to the interview stage, other things like how good you are at communicating, how enthusiastic you are and how you get along with others come into play, but no one will care about your undergrad degree there.

Is the fact that I'm objectively a second-class physicist holding me back?

If people think you're objectively second-class, it's not because of your undergrad. Maybe you are, maybe you're not, but it's all about your research and maturity now.

Should I accept that this limitation is always going to hold me back in my chosen career path, and therefore just go and change it?

Only if you want to. Right now, funding is scarce, there's a lot of competition, and it is hard to get fellowships/grants. The difficulty varies by situation--some fields have more opportunities than others, and for people who are in between established fields, like you may be, it can take time to find your place. Almost everyone has to struggle, and it's easy to get discouraged. It's how you decide to deal with it that's important. If you want to fight for your place in academia, then do it--your undergrad is not holding you back. If you want to look for an "easier" career, that's your choice.

PS If you're worried about how fit you are for academia, and it sounds like you are, you should talk to your advisors/mentors.

There are obviously a range of UK fellowships. But if you are talking about the big ones (e.g., Welcome and MRC), then getting to the interview stage is a huge deal. Don't take this the wrong way, but if you have had multiple interviews and have not gotten any fellowships, you are blowing the interview. For example, from my understanding, the Welcome Trust is willing and able to fund ever fellowship application that they interview people for. While the success rate at interview is not 100%, it is much higher than the initial stages. I am confident that rejections that happen at the interview level have nothing to do with your undergraduate education. Realistically it has nothing to do with your post graduate education either.

Again, it depends on what you are applying for, but many of these fellowships have 5+ year eligibility windows, so you have 5 years less experience than other people. I am curious as too how much your current supervisor has helped with the applications. I would be concerned that the reason that you are not doing well at interview is that you really do not own the project.