Public, reputable sources of time sync via HTTP to use where NTP is blocked?

You can use HTTP Time Protocol / htpdate. It's not a good source, not compared to NTP.
Other options include connecting a GPS-receiver or using a dialup modem.

Personally I would complain about the sysadmins to your boss. Correct time is important if you want to accurately judge HTTP cache timeouts and other time-sensitive things

ANY web server provides the date in the headers of the response, with seconds precision.

You could just access a reputable web site such as https://google.com and trust they keep their clock sane. If you want to be super-sure you can probe multiple sites.

Using https has a benefit over NTP - it is secure!

There is actually a standard in the works for getting time accurately and securely from web servers by using multiple, timed requests: http://phk.freebsd.dk/time/20151129/