Is stream irule is necessary to fix mixed content issue?

Being that the mixed content you describe is attributed to absolute URL's in the html response, an alternative to the steam profile is this ProxyPass irule that the redirect white paper hints at:

https://devcentral.f5.com/articles/rewriting-redirects

If absolute self-referencing links are embedded in the HTTP payload, you may need to build and apply a stream profile to perform the appropriate replacements. An iRule could also be used for more complex payload replacements if necessary. For the ultimate in redirect rewriting and all other things HTTP proxy, I direct your attention to the legendary ProxyPass iRule (https://devcentral.f5.com/wiki/default.aspx/iRules/ProxyPass.html) contributed to the DevCentral codeshare by Kirk Bauer (thanks, Kirk, for a very comprehensive & instructive example!

The irule has not been maintained for later versions though some users have fixed it manually (https://devcentral.f5.com/codeshare?sid=646)

The above discussion also hints at the html profile but I'm unsure if its possible in our use case.

https://support.f5.com/kb/en-us/products/big-ip_ltm/manuals/product/ltm-profiles-reference-12-1-0/3.html

I have not experimented with the above html profile solution. Its success would depend on if we successfully make a match statement with the html tags. I would have to see the payload in question to speculate further. The ProxyPass irule and the html profile could easily be tested in a lab.

Good luck and please let us know.