What version of X3 are you on, and what version of .net are you using?  X3 uses basic authentication for the SOAP web services since version 8.  I believe the way to call X3 web services from SOAP is difference from .net Core, but if you are using a standard Console/WinForms/Web application, it should work with basic authentication.

I can successfully get a list of customer using the test tool within x3 however using the same values in an equivalent call from a .net application always returns 405.

static string _x3WSUr1 = "https://x3erpv12sqlvm.local:8443/soap-wsdl/syracuse/collaboration/syracuse/CAdxWebServiceXmlCC?wsdl";
static string _publicName = "AOWSEXPORT";
static CAdxWebServiceXmlCCService _x3WebService = new CAdxWebServiceXmlCCServiceBasicAuth();
static CAdxCallContext _callContext = new CAdxCallContext();
static CAdxResultXml _resultXML = new CAdxResultXml();

static void Main(string[] args)
{
	_callContext.codeLang = "ENG";
	_callContext.poolAlias = "ADC";
	_callContext.requestConfig = "adxwss.trace.on=off&adxwss.trace.size=16384&adonix.trace.on=off&adonix.trace.level=3&adonix.trace.size=nadxwss.optreturn=1SON&adxwss.beautify=true";
	_x3WebService.Url = _x3WSUr1;
	_x3WebService.Credentials = new NetworkCredential("...", "...");
	_x3WebService.PreAuthenticate = true;
	_x3WebService.Timeout = 6000000;
	
	ServicePointManager.ServerCertificateValidationCallback = delegate { return true; }; // fix for trust relationship error

	CAdxParamKeyValue[] objectKeys = new CAdxParamKeyValue[1];

	_resultXML = _x3WebService.query(
		callContext: _callContext,
		publicName: "BPC",
		objectKeys: objectKeys,
		listSize: 100);
}

You need to add the basic authentication to the request header.  Below is a blog post with details on how to add the authentication header to the web request in .net

www.rklesolutions.com/.../5-days-sage-x3-web-services-v12-day-4

You need to add the basic authentication to the request header.  Below is a blog post with details on how to add the authentication header to the web request in .net

www.rklesolutions.com/.../5-days-sage-x3-web-services-v12-day-4

Yes I followed that guide to produce the code sample above, it looks like that will be very useful once I can successfully connect.

The example above is using the recommended custom version of CAdxWebServiceXmlCCService ( CAdxWebServiceXmlCCServiceBasicAuth )

static CAdxWebServiceXmlCCService _x3WebService = new CAdxWebServiceXmlCCServiceBasicAuth();

The custom class is correct as far as i can see. 

using ConsoleApp.X3WebService;
using System;
using System.Net;
using System.Text;

namespace ConsoleApp.App_Code
{
    public class CAdxWebServiceXmlCCServiceBasicAuth : CAdxWebServiceXmlCCService
    {
        protected override WebRequest GetWebRequest(Uri uri)
        {
            HttpWebRequest webRequest = (HttpWebRequest)base.GetWebRequest(uri);
            NetworkCredential credentials = Credentials as NetworkCredential;

            if (credentials != null)
            {
                string authInfo = "";
                if (credentials.Domain != null && credentials.Domain.Length > 0)
                {
                    authInfo = string.Format(@"{0}\{1}:{2}", credentials.Domain, credentials.UserName, credentials.Password);
                }
                else
                {
                    authInfo = string.Format(@"{0}:{1}", credentials.UserName, credentials.Password);
                }
                authInfo = Convert.ToBase64String(Encoding.Default.GetBytes(authInfo));
                webRequest.Headers["Authorization"] = "Basic " + authInfo;
            }

            return webRequest;
        }
    }
}