Found the answer myself after some trial & error.

I needed to create my custom AdminSite, and then provide my own custom admin_view, which does the redirect. Then, in the redirect, I just set the redirect_field_name to None like so:

    def admin_view(self, view, cacheable=False):
        """
        Decorator to create an admin view attached to this ``AdminSite``. This
        wraps the view and provides permission checking by calling
        ``self.has_permission``.

        You'll want to use this from within ``AdminSite.get_urls()``:

            class MyAdminSite(AdminSite):

                def get_urls(self):
                    from django.urls import path

                    urls = super().get_urls()
                    urls += [
                        path('my_view/', self.admin_view(some_view))
                    ]
                    return urls

        By default, admin_views are marked non-cacheable using the
        ``never_cache`` decorator. If the view can be safely cached, set
        cacheable=True.
        """
        def inner(request, *args, **kwargs):
            if not self.has_permission(request):
                if request.path == reverse('admin:logout', current_app=self.name):
                    index_path = reverse('admin:index', current_app=self.name)
                    return HttpResponseRedirect(index_path)
                # Inner import to prevent django.contrib.admin (app) from
                # importing django.contrib.auth.models.User (unrelated model).
                from django.contrib.auth.views import redirect_to_login
                return redirect_to_login(
                    request.get_full_path(),
                    reverse('admin:login', current_app=self.name),
                    redirect_field_name=None # <-- Set this to None to disable the "?next=" parameter.
                )
            return view(request, *args, **kwargs)
        if not cacheable:
            inner = never_cache(inner)
        # We add csrf_protect here so this function can be used as a utility
        # function for any view, without having to repeat 'csrf_protect'.
        if not getattr(view, 'csrf_exempt', False):
            inner = csrf_protect(inner)
        return update_wrapper(inner, view)