Honestly, this sounds mostly like government rule-following gone awry.
The general principle is that a contracted or wage employee may not perform work without pay. These laws were meant to protect employees from being coerced into “volunteering”, but they are often followed to stupid extremes. (For example, at a department store we were legally prohibited from telling a customer where to find something while on break.) So the base could get in serious trouble for allowing the priest to say Mass.
The easy way around this would be just to move priests around until the shutdown is over. The rules only apply to people who work for the government, not to people who are pure volunteers. Send in priests from the local diocese to say Mass as volunteers on the base, and ask the chaplains to fill in at the local parishes.
The rules are stupid, but sometimes it’s easier to work around them. I doubt there’s any malice here so much as people trying to follow all the regulations to the letter.