IIS7 URL Rewrite to router webadmin from child of default website - iis

Question

The Mission:
To access my home router's web admin site through my main website on my home server via IIS URL Rewrite/ARR. So instead of typing in http://192.168../index.cgi to get to my router, I can go to my home site http://example.com/index.html and then click on a link to go to http://example.com/router1/index.cgi.
The Journey
So far I have been able to redirect and rewrite in a number of ways. However, success only comes when I don't use a "sub-folder" in the URL. I have found that this is due to the router's website utilizing root relative links. I have tried with little success to write some outbound rules to get around this. When I use a little regex to strip that first forward slash like this
(replaces equals sign, double or single quotation mark, and a forward slash with equals and quote)
<outboundRules>
<rule name="gallifrey-out" preCondition="IsHTML">
<match filterByTags="None" customTags="" pattern="=(\"|\')\/" />
<action type="Rewrite" value="={R:1}" />
</rule>
<preConditions>
<remove name="IsHTML" />
<preCondition name="IsHTML">
<add input="{RESPONSE_CONTENT_TYPE}" pattern="^text/html" />
</preCondition>
</preConditions>
</outboundRules>
all the images appear correctly, but the links to pages are still missing the "router1" subfolder portion and therefore throw a 404 (not found) error.
I had read somewhere that the base tag would solve this. Seeing how there was no base tag for URL Rewrite to find and rewrite, I tried injecting it into the page via the style tag like so
<outboundRules>
<rule name="gallifrey-out" preCondition="IsHTML" stopProcessing="true">
<match filterByTags="CustomTags" customTags="baseInject" pattern="(.*)" />
<action type="Rewrite" value="text/css"></style><base href="http://example.com/router1/images"><style type="text/css" />
</rule>
<preConditions>
<remove name="IsHTML" />
<preCondition name="IsHTML">
<add input="{RESPONSE_CONTENT_TYPE}" pattern="^text/html" />
</preCondition>
</preConditions>
<customTags>
<tags name="baseInject">
<tag name="style" attribute="type" />
</tags>
</customTags>
</outboundRules>
Sadly this didn't work either, like .... not at all. It did inject though, here's a look at the source
Source code screenshot
Over the past few days I have looked at a ton of sites and videos on how to accomplish this. And while I have learned a lot in the process, I still can't get this done. Everyone else seem to have different way to go about it. Eventually I will go live with this on the main server and will need access to it outside my home which means the way the page is rendered to the client needs to be route-able from the public network.
Test Bench:
An extremely old Sony VIAO laptop running Win7 Ultimate with IIS7
ActionTec MI424WR rev i with Verizon/Frontier firmware
The Don't Wannas:
Don't want to open yet another port to the public on the router
Don't want to purchase yet another domain name
Don't want to purchase a wildcard SSL cert (only a single)
The Big Question
Does anyone have any idea on how to work around root-relative hyperlinks when redirecting with url rewrite?