For a similar situation, I'm considering an unvented cathedral ceiling (see references).
The main reason to vent an attic is to prevent warm moist indoor air from contacting the roof framing and decking. When the decking is below the dew point, the moisture condenses, the wood wets, and then an assortment of bad things happen.
There's another solution to keeping roofs dry: plug the air leaks between the house and the roof. If the warm moist air cannot contact the roof deck, it can't condense and wet the roofing. This is why pink insulation usually has a kraft paper backing (vapor retarder) that faces the inside the house.
When using the standard building materials of yesteryear, air and vapor sealing was all but impossible. In this brave new world of construction with Optimum Value Engineering, blower door testing, drywall sealing, taping OSB/plywood seams, and spray insulation, air sealing is no longer the stuff of science fiction.
Today, one can reliably air, vapor, and water seal a surface with closed cell spray polyurethane foam (ccSPF). The minimum depth required (by code) is based on the foam density and your climate zone. With you being situated in Climate Zone 3, I'd expect 1 inch (R-7) would suffice and more is shaving nickels off your electric bill. Since you have so little depth available, I'd consider filling the cavity with 3 inches of spray foam (R-21) in a two-step process:
- Spray foam 1" onto the decking as close to the soffits as possible. This first layer would be applied while the decking is exposed so you're likely to get uniform edge-to-edge coverage, a good air seal, and you won't have protrusions to trim off before you can drywall.
- One of:
- For less than stuffed full, spray additional insulation to the desired depth and then cover with drywall.
- To fill the cavity, hang a row of drywall, cover it with plywood, and then spray fill the cavity. The plywood will keep the expanding foam from popping your drywall fasteners. There are videos on YouTube showing a number of ways to do this.
Do read up on the building science behind this. In my jurisdiction, pulling a permit for this type of insulation retrofit is not required. It wouldn't hurt to verify that your local building code permits this. If not, showing them the relevant portions of the International Building Code might change their mind.