Authenticate with client certificate without CORS - windows

Question

I want to be able to request a distant API with my Angular 5 app.
The API server is asking a SSL-certificate in order to authenticate the client.
This certificate is distributed - by the company that provide the API - to a limited number of users, by handing them directly an USB key that contains the certificate (which is delivered by a trusted CA).
That means that I don't have access to a .pem or .crt file. Instead, the certificate is locally stored in the Windows AD (or local store) and I can't export the private key (for obvious security reason).
But the CA is also providing drivers that allow to register a certificate (if your USB key is plugged) to your favorite browser.
So I tried with the following test project:
app.component.ts
import {Component, OnInit} from '#angular/core';
import {Http, Response, RequestOptions, Headers} from '#angular/http';
import {HttpClient} from '#angular/common/http';
#Component({
selector: 'app-root',
templateUrl: './app.component.html',
styleUrls: ['./app.component.css']
})
export class AppComponent {
title = 'TEST API';
url = 'https://myAPIurl/lgcMessagerieInbox';
result = '';
constructor(private http: HttpClient) {
}
ngOnInit(): void {
}
testAPIcall(): void {
this.http.get(this.url)
.subscribe(
data => {
console.log(data);
},
error => {
console.log(error);
}
);
}
}
app.component.html
<div class="container">
<h1>
{{ title }}
</h1>
<button class="btnStandard" type="button" (click)="testAPIcall()" >TEST</button>
<div *ngIf="result">
{{ result }}
</div>
</div>
The first good news was that the browser automatically detected the fact that the API was asking a certificate and opened a windows so I could select a certificate, enter my pin code and send the http request.
But it failed because the server doesn't accept CORS requests ( https://www.html5rocks.com/en/tutorials/cors/ ).
So I though of using the local proxy provided by Angular in order to redirect the request so the browser doesn't block it because of different domain name. But then I wasn't able to pass the certificate through the proxy.
My questions are :
Is there a way to read the certificate directly on the windows local
store from my Angular app? I'm almost certain there is not if I
trust what I've been searching on internet over the past 3 days.
How can I pass a certificate through a proxy (Angular or anything else)?
If the two options above are not available, do you know a way to access the API knowing the constraints (using an other techno but it must be
cross-platform)?