Your sample looks like nanocrystalline, so there will be broadening in the peaks. There is intensity mismatch at higher angle. To reduce chi2 value you have to follow following steps which may improve chi2 value:
1. Refine isotropic peak broadening
2. Refine background carefully
3. Use input file containing XRD data in free format
You have to ask yourself what is your intention of such refinement? Lattice parameters? Structure refinement? Size and strain approximation? The absolutely best fit?
I am not sure how much experience you have but you should ask yourself (or try to find out in literature) why Chi² or R values have been introduced and what they exactly denote. You also need to realize that refinement is a mathematical technique which does not have a primary physical meaning. It is a "stupid" technique to find a approximation which is very close to the experimental signal. The best fit does not mean that this is physically the best solution since the implemented model(s) in the software and numerous parameters determine how close you can get. This means that there are possibly other solutions which are physically closer but not considered (i.e. you get some kind of fake solution). This is easy to test using two or three comparable software packages like Fullprof GSAS or MAUD. All of them are free and have generally the same approach but different code so that i would not be surprised when you will get (slightly) different results.
Inspecting your refinement, the deviations are mainly characterized by deviations at higher angles which is somehow surprising. The changing deviations indicate that either something with the structure data (atomic positions) is not perfect...or you have a considerable texture.
Nevertheless, my most important question is: What IS your problem? Only to complain about Chi² appears to me quite poor since this is only a number which refer to deviations in your difference plot. You need to think about the reasons. This number only tells you what you should (or already) see!
: Until now I believed that except of BGMN no other software has a real fundamental parameters approach for peak shape approximation. What do you understand under this term? I also did not find anything in a (older) manual of Fullprof.
wR compares deviatons with the sum of intensities, so in principal it goes from 0 to 1.
Chi² compares deviations with the uncertainty of observatons, so it goes from 1 (errors from model may not be less than observed uncertainties) to infinite.
Infinite then means either a completely wrong model, or s very low uncertainty of observations.
This happens, typically, with synchrotro radiation patterns.
To judge if refinement improves, use chi², to judge quality of profile model, use Rp, to judge crystal stru model, use R(F²)
In this pattern intensity seems to be good enough (Rexp is small), although background is not comparatively good. Compared to Rexp both Rwp & Rp are high, which results into large chi2. In the Rietveld refinement mismatch between experimental & calculated data has mainly two reasons (provided XRD pattern is free from sample preparation effects like absorption, preferred orientation etc): 1. poor structural model (atomic positions, occupancy, ADP etc) and/or 2. poor profile parameters (peak shape profile, asymmetry parameters, background type, lattice parameters etc.).
To rule out first possibility i.e. structural effects, try to use structure-less approach (LeBail fitting) and get best fit with reasonable profile mismatch. If LeBail fitting is good, then either structural model is incorrect or XRD data has sample preparation effects.
If LeBail fitting is poor, try different peak profile, background model or manual points, asymmetry limit and asymmetry parameters, anisotropic peak shape etc.