Just tried the car hire booking engine we have here with your dates and it returned no cars in La Linea!
However, if you try searching for cars in Gibraltar, but the ES option, there is a Fiesta available which comes out at about 510 Euros.
With this option, you collect the keys and the car in Gib and then drive just drive across.
It's not cheap but most of the available stock of cars will have been booked out by now.
Justin