Post #4 shows a noninverting unity gain buffer using the OPA134 / OPA2134 that has never exhibited any instability.
When replacing with the LM4562, I am wondering if you perhaps used the old tried-and-true method when upgrading to better, faster opamps: 100nF cap directly across the rails, and 22pF cap across the feedback resistor. That should cure any oscillation problems.
Oh, really? Hmmmm....how much bigger? Remember that it is supposed to be UNITY gain, is it not?