If it's not the wing or tail flexing at higher speeds (quite possible and probably the most likely answer) or being caused by too much downthrust (again possible but should be easily measured) then it may be what looks like too small a stabilizer.

If the wing section is symetrical then you can get away with a small stabilizer area but if the wing section is flat bottomed with an arced top then it may be that the wing pitching forces are overpowering the stabilizer's attempts to hold it in control. The tail area should be at least 15% of the wing area. And even that is marginal. If it's less than 15% then you may have your problem right there.