Now that I'm home, here's what the law actually says....
Screen Shot 2016-12-29 at 3.23.14 PM.png
As has already been mentioned upthread, the law is centered around process service and jurisdiction. Amtrak can do whatever functions they want, wherever they want, provided that there is someone to answer the phone and accept service in DC.
Legally, that is not exactly correct. The law requires "Washington, District of Columbia" to be the "principal office and place of business" of Amtrak. Legally, than means considerably more than a phone and someone to answer it.
The specific meaning of "principle place of business" was decided by the Supreme Court in Hertz Corp. v Friend -
130 S.Ct. 1181 (2010). Basically, the court ruled that a corporation's "principle place of business" is, in fact, the corporation's "headquarters" or "nerve center." As decided unanimously;
...We conclude that “principal place of business” is best read as referring to the place where a corporation’s officers direct, control, and coordinate the corporation’s activities. It is the place that Courts of Appeals have called the corporation’s “nerve center.” And in practice it should normally be the place where the corporation maintains its headquarters—provided that the headquarters is the actual center of direction, control, and coordination, i.e. , the “nerve center,” and not simply an office where the corporation holds its board meetings (for example, attended by directors and officers who have traveled there for the occasion).
...
Although applied to a situation different than Amtrak, it would appear that the court defined what a "principal place of business" is, and it is not just an office and a phone. Applying that definition to Amtrak and the applicable law, it sure seems that Amtrak is required by law to maintain it's "headquarters" (as commonly defined) in the District of Columbia. If Amtrak decided to move out and only maintain a minimal presence within DC and the District pressed the issue in court, my strong suspicion is that Amtrak would lose based on the Hertz Corp. v Friend precedent.