Manipulating an arbitrary-precision ContourPlot

Question

I have a function, say

minimizeme[ω_][β_][ϵ_] = ϵ^2 ω-Log[2 (Cosh[2 β]+Cosh[2 β ϵ])]/(2 β);

I want to make a high-precision dynamic ContourPlot of it using:

plottricrit[ω_] := ContourPlot[
 D[minimizeme[ω][β][ϵ], ϵ] == 0,
 {β, 0.5, 1.0},
 {ϵ, -3, 3},
 Evaluated -> True, 
 ContourStyle -> Thick, 
 RegionFunction -> Function[
    {β, ϵ}, 
    minimizeme[ω][β][ϵ] < minimizeme[ω][β][0]],
 ImageSize -> Large, 
 PerformanceGoal -> Accuracy,
 WorkingPrecision -> 60]

Manipulate[plottricrit[ω],{ω,0.217`60,0.22545`60}]

However, I keep running into a ContourPlot::precw message which tells me that

The precision of the argument function (...) is less than WorkingPrecision.

I have tried several ways to set the right precision (i.e., using With to inject a higher $MachinePrecision, using SetPrecision, using Rationalize, etc.), but I can't seem to get rid of this error. What is the right way to deal with it?

Answer 1

New method

I found that using a step value that is arbitrary precision also works:

Manipulate[plottricrit[ω], {ω, 0.217`60, 0.22545`60, 1`60*^-6}]

Old method

For reference this was my original answer, which also works but is less clean:

plottricrit[ω0_] := With[{ω = SetPrecision[ω0, 100]},
   ContourPlot[D[minimizeme[ω][β][ϵ], ϵ] == 0, {β, 0.5, 1.},
   {ϵ, -3, 3}, Evaluated -> True, ContourStyle -> Thick,
   RegionFunction -> Function[{β, ϵ}, minimizeme[ω][β][ϵ] < minimizeme[ω][β][0]],
   ImageSize -> Large, PerformanceGoal -> Accuracy, WorkingPrecision -> 60]
 ]