If we assume that a particle can be detected at a particular location, how can we do the detecting?
how would you attempt to detect an electron, at a particular location whilst in the bound state of the hydrogen atom
By shooting photons at the atom of short enough wavelength to resolve positions to the required accuracy (much less than a Bohr radius). Which would, of course, ionize the atom (with enough kinetic energy left over for the electron to make it relativistic, if my quick back of the envelope estimate is correct), so the result of your position measurement on the electron would be useless in practical terms, but it would satisfy the requirement you have stated.
I'm mainly interested to know if there are results which give backing to The Born rule.
Yes. It's details of the latest experiments I'm looking for.Doesn't any experimental result that shows probabilities equal to the squared moduli of the corresponding amplitudes support the Born rule? Which means, all of them?
used the "photoionisation microscope" to observe some nodal structures of the hydrogen atom
Yes, but how else can we look at the atom without changing it?It should be noted that this "observation" is destructive: the process of "observing" ionizes the atom and thereby destroys the structure that was being observed. So this is not the same as what one would intuitively think of as "observing" an atom, i.e., looking at it without changing it.
how else can we look at the atom without changing it?
Agreed.You can't. But the language scientists often use to describe these experiments to lay people can easily mislead people into thinking that you can. That's why I think it's important to be clear about what is actually going on. What scientists call "looking at an atom" actually means "making destructive measurements on a lot of atoms that were all prepared the same way, and using the measurement results to make a picture".