PERFECT SOLUTION BELOW

This post might be dead and I know it already has an accepted answer. However, after a couple of hours of research I found the perfect solution.

You can just subclass the PopupMenuButton and override it's handleTap method like so:

class NonDismissingPopupMenuItem<T> extends PopupMenuItem<T> {
  const NonDismissingPopupMenuItem(
      {super.key,
      super.value,
      super.onTap,
      super.enabled = true,
      super.height = kMinInteractiveDimension,
      super.padding,
      super.textStyle,
      super.labelTextStyle,
      super.mouseCursor,
      super.child});

  @override
  PopupMenuItemState<T, PopupMenuItem<T>> createState() => _NonDismissingPopupMenuItem<T, PopupMenuItem<T>>();
}

class _NonDismissingPopupMenuItem<T, W extends PopupMenuItem<T>> extends PopupMenuItemState<T, W> {
  @override
  void handleTap() {
    widget.onTap?.call(); // this override prevents popup menu to close
  }
}

This doesn't call the parent's handleTap and therefore doesn't close the PopupMenu while still enabling you to use a tap callback.

Credits to this SO post (which I wish I could've found earlier): How not to dismiss a PopUpMenuButton after selecting an item?

79110732

PERFECT SOLUTION BELOW