Get access to the Lambda alias at Startup which you can then use to load environment specific config

LambdaEntryPoint.cs

using System;
using System.Collections.Generic;
using System.Linq;
using System.Net;
using System.Threading.Tasks;
using Microsoft.AspNetCore.Hosting;
using System.IO;
using AWSBoot.Boot;
using Microsoft.Extensions.Configuration;
using Amazon.Lambda.AspNetCoreServer.Internal;
using Amazon.Lambda.APIGatewayEvents;
using Amazon.Lambda.Core;
 
namespace SecretOrange.Api
{
    public class LambdaEntryPoint : Amazon.Lambda.AspNetCoreServer.APIGatewayProxyFunction
    {
        private ILambdaContext LambdaContext;

        public LambdaEntryPoint()
            : base(AspNetCoreStartupMode.FirstRequest)
        {

        }

        [LambdaSerializer(typeof(Amazon.Lambda.Serialization.Json.JsonSerializer))]
        public override async Task<APIGatewayProxyResponse> FunctionHandlerAsync(APIGatewayProxyRequest request, ILambdaContext lambdaContext)
        {
            LambdaContext = lambdaContext;

            return await base.FunctionHandlerAsync(request, lambdaContext);
        }

        protected override void Init(IWebHostBuilder builder)
        {
            var alias = LambdaContext?.InvokedFunctionArn.Substring(LambdaContext.InvokedFunctionArn.LastIndexOf(":") + 1);

            // Do stuff based on the environment

            builder
                .ConfigureAppConfiguration((hostingContext, config) =>
                {
                    config.AddJsonFile($"appsettings.{alias}.json", optional: true, reloadOnChange: true);
                })
                .UseStartup<Startup>();
        }
    }
}

I had to change the AspNetCoreStartupMode to FirstRequest - not sure if there are any implications. Seems ok so far.

I'll leave this code here as it may help some people but note that I found AspNetCoreStartupMode.FirstRequest caused longer cold starts by a few seconds.