Not OP, but with the aforementioned schedule, there are still weekends before and after that are still very valuable. The weekend before vacation could be used to catch up on chores and errands that would be nice to get done before the vacation, plus maybe pack. Then when you get back, your coming home to a clean place and not having to deal with running around straight away. Just be ready for only two days of work. Then another weekend to get rested up before a full week of work.
Two weekends bookending two half weeks of work bookending a week off from work is very nearly the same as a week of vacation and a week of staycation for only 5 vacation days. Sure you can't go out of town for an extended period with that plan, but I'm kinda ready to go home after 4-5 days away anyway.
Idk about you, but at this rate, I don't dare try to tell them otherwise.