Think you kinda piggy-backed on a conversation that sounded like your problem, but isn't - maybe. Fist poster in this thread wanted https on certain pages in Moodle. Discussion then moved to a load balanced setup where certain pages (internal) were http and then to hacks of code - which, as you say, aren't working for ya. In your situation (depending on what that is) you may not need hacks.
So what is your setup ... load balanced? Single RH Apache box in front with backended dedicated DB server?
Your site does load and display just fine with http:// and with https:// for login only turned on it does flip to https:// just for the login with a valid cert ... and changes back to http:// (tested with guest login)
Google links to your site, however, present one with https:// which shows the wonky version.
Site does say y'all (pardon me, am in Texas, eh!) are going to be down/maintenance etc. coming up.
In RedHat apache config (httpd.conf) one could setup a redirect there (no htaccess but a true re-write rule forcing all traffic to https. Right now, would think that needs to be done when the upgrade is made to the site ... since it's gonna be down for upgrade.
And, if switching everything to https ... you'll have to turn off the https for login only, change the URL in config.php AND run the search and replace tool for any internal links that are now in the DB of the site for http://yoursite/somestuff/image.png - as an example.
Not sure there is any recommended order in which to take the site from https on login only to https and upgrade but you might consider first flipping the entire site to https first. Doing the search and replace ... checking things out to assure they work as expected under https. Then do the upgrade.
Of course, I could have the whole thing wrong. Mind explaining a little more ... or for others who might follow your path/issue, start a new discussion thread. ;)
'spirit of sharing', Ken