The central portion of the diffraction pattern (the solid disk in the middle, the Airy disk), does decrease when the central obstruction (the secondary mirror) increases, all else being equal. But the decrease is not due to the fact that fewer photons enter the system. It is purely a change in the way light is diffracted, the maximum and minimum points are placed at slightly different angles.
As the obstruction becomes bigger and bigger (all else stays the same), the minima of the diffraction pattern (the dark rings) move slightly towards the center - the whole figure appears to "shrink" somewhat. At the same time, more and more energy is being poured into the outer maxima (the outer bright rings) which become much brighter.
Points to take home:
1. The reduction of the central disk is very slight. It takes a gigantic obstruction to achieve a fairly meager reduction.
2. By the time the central disk is significantly reduced in size, the surrounding rings have become very very strong, degrading the overall performance of the system.
Even an obstruction 95% the size of the aperture does not produce a much smaller Airy disk, it just makes the outer rings extremely bright.
The overall shape and size of the diffraction pattern does not depend AT ALL on the brightness of the star. In the same telescope, all stars generate diffraction patterns of EXACTLY the same size and shape. However, faint stars appear, to the human eye, to generate smaller patterns for the simple reason that the outer part of the pattern is less bright, and hence less visible if the star is faint.
The Airy disk APPEARS to become smaller when the star is less bright (because its edge is so dark), but in reality the diameter of the first minimum (the first dark ring) remains exactly the same for all stars in a given telescope.
This is nothing new, it's just basic optics. There's nothing here to experiment or discover.
All of the above is very well illustrated on telescope-optics.net - take a look at the Point Spread Function in figure 106 (the PSF is basically a section through the diffraction pattern, indicating its brightness at different distances from the center):
The parameter "o" is the size of the central obstruction as a fraction of total aperture. Notice how the zero points of the curve (the dark rings of the diffraction pattern) move closer to center as "o" becomes smaller, but at the same time the outer maximum points (the bright rings) become taller and taller. Also notice how it takes a pretty big obstruction (0.4x the total aperture) to achieve an unimpressive reduction (maybe 20% or so) in the size of the Airy disk, at the price of a large increase (2x or more) in the brightness of the surrounding rings.
Also see here:
The phenomenon is also discussed in the book "Star testing" by H.R. Suiter.
Finally, you could download Aberrator and calculate the PSF at different sizes of central obstruction, but the software is old and quirky, so it may require some coaxing until you figure out the right song and dance to make it work.