You have to prioritize safety and health vs $600 vs trusting your Mazda.
Truthfully, if your 03 Mazda has never given you any issues and you have kept it properly maintained it should be fine. Mazda tends to have decent products.
That said... we rent cars a lot. We have never had one of our personal cars break down and we keep them in tip top shape (Mercedes)
BUT.... Hubby has been stranded more than once when he has flown to a town and headed out in a rental to make sales calls. He has had to wait on the rental car company to get a new car to him due to tire issues, engine issues, etc. This in the end cost us more money because it takes him away from working. That said, the car companies he has used and been stranded with have all been very good with follow up and perks afterwards but it is not worth it for the 2-3 hours he has lost in time and forced to reschedule appointments and flights.
He has also taken a car back to the airport to get a new one due to issues as well.
Bottom line... there are no guarantees either way that your car or a rental won't break down.
If you go with the rental, make sure you read the fine print and have unlimited miles and have "permission" to take it across state lines if you are planning on that. If you were to decided to drop it off at another location, there are fees for that as well that add up fast! Just read carefully.
Enjoy your trip!