Restore backup fails - not enough disk space

Note that the backup size does not include empty pages, but when you actually perform the restore, the data and log files will be over 200 GB, because it has to restore exactly what the source system had (including a 200+ GB log file, regardless of how full it was).

If you don't want to risk data loss, you need to correct that at the source (e.g. shrink the log file to something reasonable), take another full backup, and restore that. I'm not sure I understand why it's difficult to get a backup - this is a pretty standard operation, and should be a service provided by anyone you're paying to host SQL Server.

You should also fix the source database to either (a) be in the right recovery model or (b) take log backups more frequently. Your log file is ludicrous because you are in full recovery and never take log backups. If you need point in time recovery, start backing up your log. If you don't, switch to simple. The log file should manage itself if you have it configured correctly. If you don't, then when it gets like this, shrinking should be a one-time operation, and then you should fix the configuration so you're not doing this again next week...

DBCC TRACEON(3104) will bypass disk space checks for restore processes.