I'm not sure I've have the answer to this problem but I have some observations from my FM antennas. It is true that the placement and diameter of the gamma match can shift the impedance. I have used gamma matches on this design that ranged from 3/8 to 1 inch diameter and could always find a perfect match. On the high power models I use a large diameter gamma to accommodate the Teflon insulator. The antenna also uses a DIN connector that must be spaced out from the base of the antenna so that hardline can be used. This places a fat gamma almost parallel to the radiator and still shows no problems with matching or performance. Tilt it too close to the radials and that will mess things up quickly.
One question I have is what material is the round radial support disk made from? I'm talking about the disk with the four holes that the radials pass through. I assume this is an insulator because it can not be conductive. Also when tuning your gamma do you find the lowest VSWR with the match set at any of it's extremes? For example, is the rod all the way inserted into the lower tube? Is the tap point between the gamma and main radiator all the way at the top of the gamma rod? Does the antenna have a good VSWR on any frequency you can check?
Your radials do look too close to the radiator at the base. Bend the four large wire connectors you have holding the radials at the base so the radials maintain the same angle all the way to the loop. I don't think adding spacers here will help. It has more to do with the sharper angle in the lower part of the radials due to the bend. Your wire connectors place the base of the radials parallel to the radiator. Bending the connectors to match the desired angle of the radials will improve this. Although it still may not be the entire problem.
Making a gamma with a 36 inch 3/8 rod and 10 inch 1/2 tube should have more then enough capacitance and length to match this antenna. The tap point from rod to radiator is typically around 27 inches up from the connector bracket. If you correct the angle of the radials at the base, your radial support disk is an insulator, and your gamma is close to these specs, I'm at a loss as to why you can't match it. You can change element lengths on this antenna by feet, making the antenna completely ineffective but yet the gamma can still be tuned to provide a perfect match in nearly every case.
The spacer disk is a non-conductive plastic material. It is the end of an empty wire spool. I will fix the angle of the wire connectors holding the four radials to get a consistent angle from end to end. I am prepared to insert spacers for a more distant standout at the radial mount points, but can do the bent to proper angle fix first.
And I can make the gamma to the suggested specs.
I have noticed more SWR response at the two extremities you ask about, but so far the closest I've gotten is about 3.1:1 well above the 27.205 I'm reaching for as a low point. With some adjustments I get only from a 4___ thru 6___ SWR.
Thanks SW