If you are assigned to one base, but given a place of duty somewhere else, is there a way to be reimbursed for daily travel?

It's a Joint Travel Regulation thing. At 24 miles, it's likely considered "area of commute", hence not reimbursable. JTR doesn't care whether your orders are messed up or not. Services are prohibited from bypassing, contradicting, etc. the regulation. We've had things like carrier shift from Everett to Bremerton for some dock work. Although the commute was greater than 50 miles and JTR said to reimburse, the Navy provided bus transportation and required the sailors to use it. That was legal.