One option appears to be missing from Manual:Upgrading and maybe it is for a good reason, but I'll have to ask first.
If you don't have shell access to the server that hosts your relatively large site, you may still be able to download the files and database that belong to it. If that's the case, I would expect that you can replicate the site on your local desktop computer or whatever (e.g. using a stack that includes PHP, etc.) and do the upgrade from the command line; then, if the upgrade has completed successfully, you can upload the database and files to replace the old ones and the job is done.
Would this imaginary scenario work, or did I just paint a rosy picture that underestimates the complexity of the process? Does it matter, for instance, that the local computer runs Windows, while the remote server is Unix-based, or that the solution stack of your local computer is not as full as the remote installation?
If it does work though, it would be worth considering as an alternative to the web-based updater (which may not work) or the rather more expensive solution of migrating your site (which not everyone can afford) and so as a way out of an impasse.