There are some huge legacy systems whose dependencies on PHPs\' mcrypt
are extremely important and vital (including the data storage in database). I need to find a
Basically I think you have mentioned all possibilities and you do not have a choice. Do not downgrade to PHP 5.6 this approach has no future.
MCrypt was removed from PHP for one of the main reasons why you want to upgrade PHP: Security. The MCrypt library is not maintained anymore. Therefore installing the MCrypt extension is also a bad idea. But it can be a temporary solution (follow e.g. those instructions https://serverpilot.io/community/articles/how-to-install-the-php-mcrypt-extension.html).
The only good solution is migrating from mcrypt to something else. There are questions regarding this topic on Stackoverflow already (e.g. Upgrading my encryption library from Mcrypt to OpenSSL). Alternativly you could use some encryption library. Migrating a large amount of code/data might be a pain but this is the most future-oriented approach in this case.