Most efficient way to handle Firefox updates in computer labs with DeepFreeze?

Maybe this will not solve your problem but, if I were you, I would give a look at Firefox ESR releases:

Mozilla will offer an Extended Support Release (ESR) based on an official release of Firefox for desktop for use by organizations including schools, universities, businesses and others who need extended support for mass deployments

and

Releases will be maintained for approximately one year

We turn off updates entirely for everything except anti-virus and the stock Windows Updates on Frozen workstations, to avoid having applications download and then re-download the same data over and over every time a machine is restarted.

From here, what we used to do was go around every few months and thaw a set of machines manually, and update any known apps. This may include any of the following: Firefox, Chrome, Java, Acrobat Reader, iTunes, QuickTime, etc. We'd then fix the default profile once after those updates to clean up any nag messages like what you're seeing.

What we do now every few months instead is thaw one machine and apply and all the changes we want. This may include more than just updates. Then we use sysprep + CloneZilla to create an image of the hard drive that we will then use to rebuild other machines.

Because this leaves us with an image file that we can use to quickly restore the state of a computer as needed, as well as few other factors (Windows 7's UAC feature, better security at the network level, no admin rights for students), we are considering removing DeepFreeze all-together, and in fact have a trial in a small lab going this semester to see if it causes any problems.