Perhaps try NMaximize with constraints on the range of the parameters and Method->"RandomSearch"

As with all the other N-functions, all the variables, except those you are asking it to find, must have been assigned specific concrete numerical values before it starts.

There is always at least a dozen different ways of doing anything in Mathematica.

Do you have a reasonably good idea for the range of each parameter?

Can you write a reasonably simple expression for the magnitude of the error between a given set of parameters and the optimum?

If you could provide that sort of information then perhaps someone could try a few of the dozen different ways and see if they find one that converges at an acceptable rate.

This is really cool!

I guess the problem is it it's just slow, the fitting. Maybe it is as good as it can be, quite disappointing.

In your Module in your local variable list there is b,,Qv and that worries me.

If I insert

Print[{Qw, pw, h0, \[Sigma]0, intensity}];

immediately prior to the last line of your module, where it returns the value of intensity, and run your code then there is a vast long list of lines printed which show me that your function is being evaluated for a variety of different parameters and different values of intensities are being returned. Note: you may need to kill the math kernel to get control over the machine again because of the stream of output swamping the machine.

I was hoping that seeing the parameters and result from your int function might provide a clue.

StepMonitor just returns p1 and that's it.

 fitm = FindFit[
      datint, {k1*int[q, {p1, p2, p3, p4}, h, \[Sigma]] + c}, {{\[Sigma], 
        0.0086}, {p1, -0.487}, {p2, 1.07}, {p3, 
        0.211}, {p4, -0.1264}, {k1, (2*10^6)^-1}, {h, 17}, {c, 0.001}}, 
      q , Method -> "LevenbergMarquardt", StepMonitor -> Print[p1]]

Tiny probable typo:

dat3 = Import["Non-annealed data.txt", "Data"];
fitm = NonlinearModelFit[datint,...

I'm guessing datint should be dat3. True?

Any other typos if you carefully look over the files you posted?

Thanks