Hi j bg,

Looks like a bug, you should report it to Wolfram Support. It seems to be caused by the small y values. Here is a workaround.

Scale the y values by 10^13

measScaled = MapAt[#*10^13 &, meas, {All, 2}];

Plot with scaled frame ticks

mwePlot = ListLogLogPlot[measScaled,
  GridLines -> Automatic,
  GridLinesStyle -> Black,
  FrameStyle -> Black,
  Frame -> True,
  FrameLabel -> {"PAS Detuning [GHz]", 
    "\!\(\*SubscriptBox[\(K\), \(\[Nu]\)]\)/I \
[\!\(\*SuperscriptBox[\(cm\), \(5\)]\)\!\(\*SuperscriptBox[\(mW\), \
\(-1\)]\)\!\(\*SuperscriptBox[\(s\), \(-1\)]\)]", "", ""},
  BaseStyle -> {FontSize -> 14},
  PlotLegends -> \
{"\!\(\*SuperscriptBox[\(\[InvisiblePrefixScriptBase]\), \(86\)]\)Sr, \
exp. "},
  PlotRange -> {{3, 500}, {10^-3, 10^2}},
  PlotStyle -> {GrayLevel[0.5], PointSize[Large]},
  ImageSize -> Large,
  FrameTicks -> {{Charting`ScaledTicks[{Log[# 10^13] &, Exp[# ]/10^13 &}], Automatic}, 
      {Automatic, Automatic}}]