If you are not comfortable using the mysqldump command line tool, here are two GUI tools that can help you solve this problem, although you should be able to upload them to the server via FTP!
Adminer is a thin and very efficient DB Manager tool that is at least as efficient as PHPMyAdmin and has only ONE ONE FILE that must be uploaded to the server, which makes it extremely easy to install. It works better with large tables / databases than PMA.
MySQLDumper is a tool designed specifically for exporting / importing large tables / databases, so it will not have a problem with the situation you are describing. The only dowside is a bit tedious to install, as there are more files and folders (~ 350 files in ~ 1.5 MB), but there should be no problems downloading via FTP, and it will definitely get the work done :)
So, my advice: try Adminer first, and if this one doesn't work either, go to the MySQLDumper route.
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