Try checking the Mousesavers page for car rental codes that may work for you.
Not quite understanding the logic about renting a car for better gas mileage. Do you anticipate saving enough on gas to pay for the rental?
If you are a member of AAA, they have some pretty decent rates. The promotions they have also get better the closer you get to the dates you are going, so make sure to check back, even if you are already reserved, to see if a better promotion is applicable! Good luck!!
You'd be saving about 3 mpg according to fueleconomy.gov numbers. You'll save about $25 on fuel for every 1000 miles driven. Additional space is one thing but renting a car to save on gas almost never pans out. It does sound good when justifying the idea to a spouse.
And the SH-AWD model gets even worse fuel economy and has a slightly smaller trunk to boot 
So ya it's doable, but you do have to get creative with packing, basically a large number of small bags versus a small number of big ones. We have those enormous Britax Advocate car seats in the back and cram a bunch of stuff in the space between them and the front seats. Having the valet wait for us to unload everything is always amusing, they have this look on their face like "you fit THAT into THAT???" 
That leaves plenty of time to find a discount then. Still check those tires if you go through SJ airport, though, I honestly can't believe they rent cars out like that, it's a real danger and I'm not the type who's usually very paranoid about this sort of thing...Hey we (two adults and two 2-year olds, one disabled so we have to take his wheelchair too) have taken our 2009 TL SH-AWD to Disneyland 4x already ... it's headed down again tomorrow
Ok kidding aside, if you're dead set on renting a car, I travel a lot for work and find Budget and Dollar at SJ airport to have the best rates in the area. I agree with the suggestion to check around because rates always change. One thing though ... if you rent from SJ airport, check your rental car's tires very carefully. Those ramps (the ones with the spikes) you drive over when entering and exiting the facility leave cuts on the tires. I'm a car nut and have noticed little slashes on the tires of many rental cars. Whenever I see it, I always request a new car to mitigate the risk of a blowout on the 5. The rental car places know it, too, the employees have even told me that they've lost their own tires to those ramps, but since most rentals go to business travelers who couldn't care less, no one ever does anything about it (okay, that and they're mandated by the airport or something along those lines is what I've been told). For taking your family down to So Cal, be sure to look over the tires carefully, I've made the drive between the Bay and LA probably about 200+ times in my life and have suffered 2 blowouts, and trust me, it's not fun...
Ok starting to blab now, but one last thing ... EVERYTHING is negotiable. Oftentimes if you call the rental car place and just sweet talk a bit, you can get a discount. It's harder for holiday weekends but SJ airport has so many rentals sitting around, inevitably someone will have excess inventory, especially at the last minute.
I'm a light packer since I do a lot of business travel (okay plus I'm a dude and allowed to be smellier right?!?!) and can get by on a single backpack for a week easily so long as weather conditions remain constant, which of course we know they will because it's summer in California Getting a bigger car will definitely lower the pressure to cram cram cram, and of course allow more space to bring back souvenirs and what not for little ones (okay I guess that goes for your own souvenirs too hehe).
I mean, I was thinking about saving my car from the damage of rocks etc., but I completely forgot that when we leased the car we picked the lowest mileage (10K or 12K?) ...something like that - and I definitely don't want to end up paying overage later!
