Changing "the" to "a" would be correct grammar, but would change the meaning. The writer is urging the reader to book now, and the phrase "a last-minute rush" would mean the reader rushing. That would be written as "Book now and avoid rushing at the last minute." or "Book now and avoid a last-minute rush."
Here "the" is correct, and "the last-minute rush" does not talk about the reader rushing.
For popular events, it is normal and expected that a rush of bookings (very many in a short time) will happen just prior to the start of the event (a "last-minute rush" of bookings for that event). This sentence is talking about booking one specific event. That event will have one last-minute rush of bookings. Both writer and reader know which one last-minute rush is referred to by "the last-minute rush."