Holes in ElementMesh with ToElementMesh of ImplicitRegion

I hope I interpreted your question correctly that you want a more accurate ElementMesh representation of the region.

First we create a high quality Graphics of the region of interest.

isovalue = 1.29897233417072;
(* Add some margins to plot range to get connected region. *)
tolerance = 0.05;
plot = ContourPlot[
  f[x, y, 4],
  {x, y} ∈ Cuboid[{-0.5, -0.5} - tolerance, {0.5, 0.5} + tolerance],
  Contours -> {isovalue},
  ColorFunction -> GrayLevel,
  (* We need high quality plot for ImageMesh later. *)
  PlotPoints -> 200,
  Frame -> None
]

Create MeshRegion from Graphics object.

mreg = ImageMesh[ColorNegate[plot]]

And convert it to ElementMesh.

Needs["NDSolve`FEM`"]
mesh = ToElementMesh[mreg,"MeshOrder"->1]
(* ElementMesh[{{7., 353.}, {7., 353.}}, {TriangleElement["<" 1057 ">"]}] *)

mesh["Wireframe"]

Another approach is:

reg = ToElementMesh[
   ImplicitRegion[
    f[x, y, 4] < isovalue && {x, y} \[Element] cell, {x, y}], 
   "MaxBoundaryCellMeasure" -> 0.01, MeshQualityGoal -> 1, 
   PerformanceGoal -> "Quality", MaxCellMeasure -> 0.01, 
   "BoundaryMeshGenerator" -> {"RegionPlot", "SamplePoints" -> 41}];

reg["Wireframe"]

One thing to be a bit careful about is the question if the holes intersect the boundary. From the mesh it does not look like it but the math might say it.