Don't have the time to fully answer this yet, but here's a debug toolsa debug tool I developed previously. First off we can figure out how OBJ is exported
getFormatExportData["OBJ"]
{"FormatName" -> "OBJ", "DefaultElement" -> "Graphics3D",
"DocumentedElements" -> None, "Function" -> System`Convert`OBJDump`ExportOBJ,
"FunctionChannels" -> {"Streams"}, "HiddenElements" -> None,
"Options" -> {"InvertNormals", "VerticalAxis", "Comments"},
"Sources" -> \
{"/Applications/Mathematica.app/Contents/SystemFiles/Kernel/SystemResources/\
MacOSX-x86-64/Convert/Common3D.mx",
"/Applications/Mathematica.app/Contents/SystemFiles/Kernel/SystemResources/\
MacOSX-x86-64/Convert/OBJ.mx"}, "BinaryFormat" -> True,
"File" -> "/Applications/Mathematica.app/Contents/SystemFiles/Formats/OBJ/\
Export.m"}
Then we get the defs for that func (you may need to load the .mx first)
System`Convert`OBJDump`ExportOBJ // GeneralUtilities`PrintDefinitionsLocal
ExportOBJ[strm_, expr_, opts___] := Block[
{res, nopts, method, a, expr1, obj},
nopts = Flatten @ {opts};
method = ("ImportExportMethod" /. nopts) /. {"ImportExportMethod" -> $ImportExportMethod};
nopts = DeleteCases[nopts, "ImportExportMethod" -> _];
a = Flatten[expr /. {RuleDelayed -> List}];
nopts = Union @ Flatten @ Join[nopts, Most @ a];
nopts = Select[nopts, Function[Equal[Head @ #, Rule]]];
obj = Last @ a;
expr1 = "Graphics3D" -> obj;
res = dispatchExportOBJ[method, strm, expr1, nopts];
res /; UnsameQ[res, $Failed]
];
Which means it's handled by dispatchExportOBJ and so looking at those defs we get back to either System`Convert`OBJDump`LegacyExportOBJ or System`Convert`OBJDump`ImprovedLgacyExportOBJ, the latter of which seems to delegate to GeometryTools`WGTAssimpExport and the former of which is just a bunch of WriteString calls which could be modified to have higher precision.
Can't get you to a full solution at this point in time though.
