Is MathieuC for moderately large imaginary arguments broken?

Bug introduced in 3.0 and persisting through 12.0
(reported as CASE:3208982)

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I’m trying to plot MathieuC[-3,0.3,I x] for $xin[0,10]$, and here’s what I get even with arbitrary precision arithmetic (here I use ListPlot and Table instead of Plot to make computation faster and use a regular grid):

$MaxExtraPrecision=200;
ListPlot[Table[{z,N[Re@MathieuC[-3,3/10,I z],50]},{z,0,7,1/100}],
  PlotRange->{-0.6,0.6}, Joined->True]

So, starting from about z==3.3, the results are unreliable at all. Comparing machine precision and arbitrary precision computations with N[], I get this:

N[Re@MathieuC[-3, 3/10, 5 I]]
N[Re@MathieuC[-3, 3/10, 5 I], 50]

-1.26013246174486*10^28

-1.2601324617438073657004964476674284869363635740962*10^28

So, it seems not even lack of working precision — looks like the algorithm is broken. To make sure I’m not misunderstanding the supposed behavior of Mathieu functions, I’ve also checked it by solving Mathieu equation numerically:

With[{a = -3, q = 0.3},
 sol = NDSolveValue[{
      -w''[z] - 2 q Cosh[2 z] w[z] == -a w[z],
      w[0] == Re@MathieuC[a, q, 0],
      w'[0] == 0
      }, w, {z, 0, 7}, MaxSteps -> 10^6]
 ];
$MaxExtraPrecision=200;
Show[
 Plot[sol[z],{z,0,7}, PlotStyle->Darker@Green, PlotPoints->300],
 ListPlot[Table[{z,N[Re@MathieuC[-3,3/10,I z],50]},{z,0,7,1/100}],Joined->True],
PlotRange->{-0.6,0.6}]

This shows that the function indeed should look nicer, but apparently MathieuC can’t calculate it for even moderately large imaginary arguments. It also appears that Mathematica versions 5 to 12 give exactly the same results (sometimes with differences in several least significant digits).

Is it a bug, or is it documented somewhere? Are there any workarounds, allowing me to evaluate Mathieu functions not as slowly as via NDSolve, and still not reimplementing them like e.g. here?