1. The host doesn't allow SSH access.
2. I can't install outside the webtree, because the host is using PHP's open_basedir restriction. In their case, nothing outside the web directory can be read by PHP.
3. I tried using the PEAR 1.4 remote installation instructions involving Net_FTP, but for some reason my local installation was not being mirrored remotely. If there's an error log describing what went wrong, I couldn't find it.
4. Using the web-based installation with go-pear.php isn't supported under PEAR 1.4 right now. I tried it anyway, and it worked until the package files had to be decompressed. PEAR wasn't able to create a temporary directory in which to extract the files.
5. The PEAR manual no longer contains the manual installation instructions. Apparently the directory structure has become more complicated in recent versions, and simply downloading the .tar.gz files of the various packages is not a Good Idea.

All in all, it wasn't a very pleasant experience, but at least now the website is working with our own copy of the files. The web host we're using is fine, but their setup is hamstrung, in my opinion, by their over-reliance on the control panel software driving their shared hosting. I could fix this problem easily myself, but I don't have access to the configuration files.