TL;DR: I don't think English truly has an equivalent, possibly due to business/government culture: I would suggest that "word-by-word" translation is not appropriate in this case, but rather that the whole section rephrased in a more typically Anglo-phonic way.
In general, this will very not only between countries but between ministries/departments/agencies.
Most governmental and corporate responses that I have received in English do not mention what department handled the receipt of the message. Generally, they either:
1) used a sentence fragment, especially on a form (e.g. "Received date: Oct 18, 2018"(because I live in the US, where we insist on using confusing dates and measurement systems) or "Timestamp: 2018-10-18 15:40")
2) if mentioned in a letter, the name of the organization or department that ultimately received it (e.g. "Your submission was received by the Claims Reimbursement Department on 12 October 2012). This is analogous to Tᴚoɯɐuo's mention of Court Clerks in the US system.
3) if mentioned in a letter, the pronoun "we" is sometimes used. E.g. "We have received your submission on 12 October 2012".
Concerning why I wouldn't use any of the translations you were provided with:
"Mail room" is technically a physical space where mail is delivered to by external mailing services and sorted, and from where it is then delivered to individuals and department by internal employees. It is also used to collectively refer to those internal employees who sort and do the "last mile/foot" delivery of packages and physical mail. I would not expect to see this used in official communication in any but the rarest of official communications.
"Mail and Messaging Department" is something that I have never heard, but I would assume it to be the official name for the workers. In a partially large agency (that had multiple physical office locations), it could also be in charge of resending physical mail to specific locations. Like mail room, I would not expect it to be referred to in most offical communications.
"Dispatch office" and "dispatch service" generally refer to outgoing messages, but depending on the agency structure, could handling incoming messages as well.