How can a Logic App workflow be configured to use private DNS resolution with Azure Bicep?

Question

I have been assigned to utilize a private endpoint within a virtual network through an Azure Logic App deployed in the same network. I successfully deployed the workflow and integrated it with the virtual network using the following:

resource logicAppVnetIntegration 'Microsoft.Web/sites/networkConfig@2024-04-01' = {
  parent: logicApp
  name: 'virtualNetwork'
  properties: {
    subnetResourceId: resourceId('Microsoft.Network/virtualNetworks/subnets', vnetName, logicAppSubnetName)
    swiftSupported: true
  }
}

Additionally, I have configured the following Logic App settings:

appSettings: [
  {
    name: 'WORKFLOWS_MANAGED_IDENTITY'
    value: 'SystemAssigned'
  }
  {
    name: 'WEBSITE_VNET_ROUTE_ALL'
    value: '1'
  }
  {
    name: 'WEBSITE_DNS_SERVER'
    value: '168.63.129.16'
  }
]

However, when I send a request to the private endpoint, I receive the following error:

{
  "error": {
    "code": "403",
    "message": "Public access is disabled. Please configure private endpoint."
  }
}

Here is the visualization of my bicep code:

After debugging, I suspect that the issue is due to the Logic App failing to resolve the private endpoint. If anyone has encountered a similar issue, I would appreciate any guidance. Bicep templates would be especially helpful.