You can file a petition with the FCC to have your landlord's rule challenged BEFORE installation:
Q: How do I file a petition or request a waiver at the Commission?
A: There is no special form for a petition. You may simply describe the facts, including the specific restriction(s) that you wish to challenge. If possible, include contact information such as telephone numbers for all parties involved, if available, and attach a copy of the restriction(s) and any relevant correspondence. If this is not possible, be sure to include the exact language of the restriction in question with the petition. General or hypothetical questions about the application or interpretation of the rule cannot be accepted as petitions. To file a Petition for Waiver, follow the requirements in Section 1.4000(c) of the rule. The local government, community association or landlord requesting the waiver must demonstrate "local concerns of a highly specialized or unusual nature."
Petitions for declaratory rulings and waivers must be served on all interested parties. For example, if a homeowners' association files a petition seeking a declaratory ruling that its restriction is not preempted and is seeking to enforce the restriction against a specific resident, service must be made on that specific resident. The homeowners' association will not be required to serve all other members of the association, but must provide reasonable, constructive notice of the proceeding to other residents whose interests foreseeably may be affected. This may be accomplished, for example, by placing notices in residents' mailboxes, by placing a notice on a community bulletin board, or by placing the notice in an association newsletter. If a local government seeks a declaratory ruling or a waiver from the Commission, the local government must take steps to afford reasonable, constructive notice to residents in its jurisdiction (e.g., by placing a notice in a local newspaper of general circulation). Proof of constructive notice must be provided with a petition. In this regard, the petitioner should provide a copy of the notice and an explanation of where the notice was placed and how many people the notice reasonably might have reached.
Finally, if a person files a petition or lawsuit challenging a local government's ordinance, an association's restriction, or a landlord's lease, the person must serve the local government, association or landlord, as appropriate. You must include a "proof of service" with your petition. Generally, the "proof of service" is a statement indicating that on the same day that your petition was sent to the Commission, you provided a copy of your petition (and any attachments) to the person or entity that is seeking to enforce the antenna restriction. The proof of service should give the name and address of the parties served, the date served, and the method of service used (e.g., regular mail, personal service, certified mail).
All allegations of fact contained in petitions and related pleadings before the Commission must be supported by an affidavit signed by one or more persons who have actual knowledge of such facts. You must send an original and two copies of the petition and all attachments to:
Secretary, Federal Communications Commission
445 12th Street, S.W.
Washington, D.C. 20554
Attention: Media Bureau
A properly filed petition with the required notifications may be enough to get your landlord to be more reasonable. The petition may cause them to consult an attorney, who might point out that some of their restrictions are illegal and that there is a high probability there will be a ruling against them. The size restriction is especially weak in my view. I would not be surprised if a kickback from a local DishNetwork installer was involved.
Your situation is somewhat amusing to me. I have seen apartment buildings in Atlanta, Georgia where very sturdy exterior satellite antenna mounts are used on all kinds of walls. The antennas are positioned 20, 30, 40, and 50 feet above driveways, walkways, and other open areas. They are cantilevered in place. The cable access holes are drilled through all kinds of materials. Usually the installer puts a sleeve in the hole, puts weatherproof grommets on both sides, and seals the installation up with outdoor weather resistant sealant. There is not going to be any water, bug, or other intrusion through the installation holes. Yes, the antenna could get torn off the side of the building, during a F4 or F5 tornado. I guess when landlords hate the local cable company's TV offerings and service as much as the tenants, they allow innovative but safe satellite antenna installations.