I would not know how to use a checksum for this, especially since with some settings it could happen that files are changed slightly when uploaded. A relatively safe way (and faster than uploading individual files) would be to upload a zip, check the checksum of that and then unzip it. But if there was an error transferrig a zip, it would probably be broken and not unzip at all.
If you don't have the option to unzip yourself, a good hoster will do it for you. Same if you download, they can create a zip of your installation for you.
If you have access to the backend, you can also use a backup extension. Then download the backup and extract it locally. There are also a few php scripts to create compressed backups.
Anyway, there are still often good reasons to compare folders. I use PhpStorm for that. Good ftp programs, such as WinSCP, also have folder compare options. Will take a long while. To be really sure, you can (with PhpStorm, at least) even compare content. For a whole Joomla installation, that can take all day, though, even with a fast connection.
Sorry, I see you asked about FileZilla. I have a strong dislike for FileZilla, so have not used it for years. If it does not have folder comparison, dump it.
Comparing sizes would be easy if you had ssh access to the server. But you probably don't - then I would not know how you could do that. See View the size of a directory with filezilla which links to this: https://forum.filezilla-project.org/viewtopic.php?t=34641