The Federal Service Labor-Management Relations Statute (FSLMRS) that you've linked in your question actually spells it out when it comes to what federal employees can legally do when it comes to collective bargaining. "The Statute" came about after the passing of the Civil Service Reform Act (CSRA) of 1978 under the Carter administration.
Under the CSRA, the strike was taken away from federal employees. The new legal framework codified similar collective bargaining rights for federal employees as private sector workers have under the NLRA, but like a lot of US labor history the establishment of a cohesive legal framework for labor relations removed key mechanisms that have been traditionally used throughout US history to disrupt production.
Federal employees are entitled to form and/or affiliate with a union and, by law, the government must negotiate with the union the workers elect to be represented by. The union then becomes the sole representative of the workers' 'bargaining unit'. Through unions like the National Federation of Federal Employees (NFFE), the National Association of Government Employees (NAGE), and the American Federation of Government Employees (AFGE), union officials negotiate with management on issues of personnel policies, practices, and matters that affect working conditions. Employees cannot negotiate around anything that is "specifically provided for by Federal statute" (page 3). This means that issues around wages/salaries, retirement benefits, and healthcare aren't negotiable through the union.
Beyond standard negotiations, federal workers, through their representation, may file a case with the Federal Labor Relations Authority (FLRA)--the US government's internal version of the National Labor Relations Board (NLRB). If the government violates any of the rights laid out under the FSLMRS, the worker(s) may file an Unfair Labor Practice (ULP). Conversely, if the union violates any of the management rights outlined in the FSLMRS, a ULP may be filed against them instead. This process takes a great deal of times and has to escalate several times before it reaches an FLRA Administrative Law Judge (ALJ). Without going into too much detail, the judge decides whether or not a complaint has merit and decides what relief, if any, is granted.
Now, legally speaking, that's all the leverage that federal employees have. The sickout method you've described above is a method of direct action, but it isn't legally protected. This means that actions like these have to be worker-led with the workers potentially facing penalization. Even picketing under the FSLMRS is restricted to 'informational' picketing (pages 26, 46)--a form of protest that may be utilized so long as it doesn't impact the operations of the agency. Even then, the law aims to prevent what's known as 'self-help' tactics until all FLRA measures have been exhausted. If a union tried to authorize any direct action (like a sickout), they would likely have an ULP filed against them and the FLRA has historically ruled in favor of the government.
If you'd like more information on what it looks like for federal workers, the AFGE provides a comprehensive look at the collective bargaining process from their end.
For more information on the history of direct action as used by labor, the Industrial Workers of the World have a storied history of it. libcom.org also provides a guide to worker-led direct action.
