The gameplay is not protected by copyright. There are many shooters, they are different, and there is nothing wrong with starting to make another game in the shooter genre.
So in our case, BigCompany LTD can't have any claims about the genre being the same in your games. A reasoned response to this claim in a reply letter could be:
The similarity of the gameplay loop and the mechanics of the game is not a violation of copyright and represents the realization of the freedom of creativity enshrined in the US Constitution, section 17 U.S. Code §102 b): «In no case does copyright protection for an original work of authorship extend to any idea, procedure, process (A "device", "machine", or "process" is one now known or later developed), system, method of operation, concept, principle, or discovery, regardless of the form in which it is described, explained, illustrated, or realized in such work».
US courts also do not support the idea of copyright protection of the mechanics. For ex.: Tetris Holding, LLC v. Xio Interactive, Inc., 863 F. Supp. 2d 394, 403 (DNJ 2012) ("When similar features in a videogame are as a practical matter indispensable, or at least standard, in the treatment of a given idea, they are treated like ideas and are therefore not protected by copyright") (internal citations omitted).
Gameplay elements of a video game are ineligible for copyright. Gameplay concepts fall into the idea-expression distinction that had been codified in the Copyright Act of 1976, in that copyright cannot be used to protect ideas, but only the expression of those ideas. The United States Copyright Office specifically notes: "Copyright does not protect the idea for a game, its name or title, or the method or methods for playing it. Nor does copyright protect any idea, system, method, device, or trademark material involved in developing, merchandising, or playing a game. Once a game has been made public, nothing in the copyright law prevents others from developing another game based on similar principles." Courts also consider scènes à faire (French for "scenes that must be done") for a particular genre as uncopyrightable. It is generally recognized in the video game industry that borrowing mechanics from other games is common practice and often a boon for creating new games, and their widespread use would make them ineligible for legal copyright or patent protection.