While this action may indeed cause the DoD to blacklist Anthropic from doing business w/the government, they probably were being as careful as they could be not to double down on the nose-thumbing.
I don't think it's addressed to Hegseth, but to anyone who might be sympathetic to Hegseth. Which I think actually strengthens your point, the goal appears to be to make it so the only possible complaint with the letter for someone sympathetic to the administration is "but mass domestic surveillance / fully autonomous weapons are legal" and not "look at this lunatic leftist who calls it the department of defense".