URL rewrite loses Virtual Directory prefix using JavaScript root-based urls - redirect

Question

We have some third-party service on http://localhost:1234/, and we want to be able to access it externally via https://my-site.com/MyService/
For that we have IIS with Virtual Directory (Application) MyService and URL rewrite rule
<rule name="my-service" enabled="true" patternSyntax="Wildcard" stopProcessing="true">
<match url="*" />
<action type="Rewrite" url="http://localhost:1234/{R:0}" />
</rule>
This service renders html with some root urls like Page 1
But following such links will lead us to https://my-site.com/MyService/page1.html
In order to fix it we have another URL rewrite outbound rule
<rule name="ReverseProxyOutboundRule1" preCondition="ResponseIsHtml1">
<match filterByTags="A, Form, Img, Link, Script" pattern="^/(.*)" negate="false" />
<conditions logicalGrouping="MatchAll" trackAllCaptures="true" />
<action type="Rewrite" value="https://my-site.com/MyService/{R:1}" />
</rule>
And it works fine and renders absolute urls Page 1
However, if we have JavaScript
// redirect
location.href = "/page2.html";
// AJAX
var xhr = new XMLHttpRequest();
xhr.open("GET", "/page2.html");
xhr.send();
// fetch
fetch("/page2.html");
those root-based urls will be resolved as https://my-site.com/MyService/page2.html
As this service is third-party, I can't just go to modify all those JavaScript code to include /MyService prefix. And those urls such as /page2.html are constructed dynamically so it's impossible to just use URL rewrite module to fixup corresponding JavaScript.
I was trying to add URL rewrite to the root website to capture such urls via http referer
<rule name="MyService" stopProcessing="true">
<match url="^(?!MyService).+$" />
<conditions>
<add input="{HTTP_REFERER}" pattern="https://my-site.com/MyService*" />
</conditions>
<action type="Redirect" url="https://my-site.com/MyService/{R:0}" redirectType="Found" />
</rule>
but it still doesn't work properly and behaves very fragile.
Is there a way to solve such supposedly very common task easily?