With this informations I tried with this pins without success :
#define INT_GPIO 35
#define MOSI_GPIO 23
#define MISO_GPIO 38
#define SCK_GPIO 18
#define CS_GPIO 33

I think there must be weird things happening in the SPI initialization with M5.begin() and all that, in any case this code doesn't work:

#include <Arduino.h>
#include <M5Unified.h>
#include <M5Module_LAN.h>
#include <SPI.h>

#define _ASYNC_WEBSERVER_LOGLEVEL_ 2
M5Module_LAN LAN;

//////////////////////////////////////////////////////////

// Optional values to override default settings
// Don't change unless you know what you're doing
#define ETH_SPI_HOST SPI3_HOST
#define SPI_CLOCK_MHZ 25

// Must connect INT to GPIOxx or not working
#define INT_GPIO 35
#define MOSI_GPIO 23
#define MISO_GPIO 38
#define SCK_GPIO 18
#define CS_GPIO 33

//////////////////////////////////////////////////////////

#include <AsyncTCP.h>
#include <AsyncWebServer_ESP32_W5500.h>

AsyncWebServer server(80);

const char *PARAM_MESSAGE = "message";

void notFound(AsyncWebServerRequest *request)
{
  request->send(404, "text/plain", "Not found");
}

void setup()
{
  M5.begin();
  Serial.begin(115200);

  if (esp_netif_init() != ESP_OK)
  {
    Serial.println("main::setup(): Error: Unable to initialize network interface");
  }

  uint8_t cs_pin = 33;
  uint8_t rst_pin = 0;
  uint8_t int_pin = 35;

  SPI.begin(SCK, MISO, MOSI, -1);
  LAN.setResetPin(rst_pin);
  LAN.reset();
  LAN.init(cs_pin);

  Serial.print(F("\nStart AsyncSimpleServer_ESP32_W5500 on "));
  Serial.print(ARDUINO_BOARD);
  Serial.print(F(" with "));
  Serial.println(SHIELD_TYPE);
  Serial.println(ASYNC_WEBSERVER_ESP32_W5500_VERSION);

  Serial.println("Default SPI pinout:");
  Serial.println("SPI_HOST:" + String(ETH_SPI_HOST));
  Serial.println("MOSI:" + String(MOSI_GPIO));
  Serial.println("MISO:" + String(MISO_GPIO));
  Serial.println("SCK:" + String(SCK_GPIO));
  Serial.println("CS:" + String(CS_GPIO));
  Serial.println("INT:" + String(INT_GPIO));
  Serial.println("SPI Clock (MHz):" + String(SPI_CLOCK_MHZ));
  Serial.println("=========================");

  ///////////////////////////////////

  // To be called before ETH.begin()
  ESP32_W5500_onEvent();

  // bool begin(int MISO_GPIO, int MOSI_GPIO, int SCLK_GPIO, int CS_GPIO, int INT_GPIO, int SPI_CLOCK_MHZ,
  //            int SPI_HOST, uint8_t *W5500_Mac = W5500_Default_Mac);
  // ETH.begin( MISO_GPIO, MOSI_GPIO, SCK_GPIO, CS_GPIO, INT_GPIO, SPI_CLOCK_MHZ, ETH_SPI_HOST );
  Serial.print("Network::connectEthernet(): Connecting to Ethernet with DHCP ");
  byte mac[] = {0xDE, 0xAD, 0xBE, 0xEF, 0xFE, 0xED};
  if (!Ethernet.begin(mac))
  {
    Serial.println("Network::connectEthernet(): DHCP failed, aborting");
  }

  EthernetLinkStatus linkStatus;
  while ((linkStatus = Ethernet.linkStatus()) != LinkON)
  {
    Serial.print(".");
    delay(100);
  }

  Serial.println(" ...Connection established!");
  Serial.println("Network::connectEthernet(): IP address   : " + LAN.localIP().toString());
  Serial.println("Network::connectEthernet(): Subnet mask  : " + LAN.subnetMask().toString());
  Serial.println("Network::connectEthernet(): Gateway IP   : " + LAN.gatewayIP().toString());
  Serial.println("Network::connectEthernet(): DNS          : " + LAN.dnsServerIP().toString());
  ESP32_W5500_waitForConnect();

  ///////////////////////////////////

  server.on("/", HTTP_GET, [](AsyncWebServerRequest *request)
            { request->send(200, "text/plain", "Hello, world from AsyncSimpleServer_ESP32_W5500"); });

  // Send a GET request to <IP>/get?message=<message>
  server.on("/get", HTTP_GET, [](AsyncWebServerRequest *request)
            {
    String message;

    if (request->hasParam(PARAM_MESSAGE))
    {
      message = request->getParam(PARAM_MESSAGE)->value();
    }
    else
    {
      message = "No message sent";
    }

    request->send(200, "text/plain", "Hello, GET: " + message); });

  // Send a POST request to <IP>/post with a form field message set to <message>
  server.on("/post", HTTP_POST, [](AsyncWebServerRequest *request)
            {
    String message;

    if (request->hasParam(PARAM_MESSAGE, true))
    {
      message = request->getParam(PARAM_MESSAGE, true)->value();
    }
    else
    {
      message = "No message sent";
    }

    request->send(200, "text/plain", "Hello, POST: " + message); });

  server.onNotFound(notFound);

  server.begin();
}

void loop()
{
  delay(1000);
}

Start AsyncSimpleServer_ESP32_W5500 on M5Stack Core2 with ESP32_W5500
AsyncWebServer_ESP32_W5500 v1.6.4 for core v2.0.0+
Default SPI pinout:
SPI_HOST:2
MOSI:23
MISO:19
SCK:18
CS:5
INT:4
SPI Clock (MHz):25