They can send test messages to the mailboxes they monitor to try to determine if your site can be used as a spam gateway. This is my suggestion.
Another possibility (as already mentioned) is that they try to improve the reputation of the IP address by sending apparently good messages from the server.
As for their blocking, their bots are usually very stupid. They do not parse HTML correctly, and they certainly do not read the form or do not support CSS. Add a field called "email" inside the display = 'none' element. Make sure this field is blank from legitimate respondents, and the bots will almost certainly fill it in anyway. Rename your real email field to another.
Then any submit form with a value in an empty field is definitely spam and can be ignored.
Other options include using Javascript (which are usually not executed) to fill in the hidden field, but this imposes more requirements on legitimate user agents (however, better than intercepting).
Provided that the technique you use is not too common or similar to those used by many other sites, the bots will not have a workaround for it.
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