That exhaust gas ‘cure’ could result in a car that stinks to high heaven! Modern engine don’t put out much CO, anyway. If you want a cleaner version of that, buy some frozen CO2 and put it in your car. Might work.
If you have time to leave the car to sit for a day or two, I’d say dust the whole interior with Delta Dust insecticide. It’ll kill pretty much any invertebrate that’s in your car. Wear a mask and protective clothes while applying it - it’s not super toxic to humans but it can make you itch/burn and other unpleasantness.
Then take a good strong shop vac to the entire car, especially including any crevices that don’t normally get disturbed, like under the seats and dash. Don’t forget between the seat pan and the seat back, and the map pockets if equipped.
And finally, I’d take it to a detail shop and have the interior fully detailed - that should wipe up any pesticide that you may have missed with the shop vac.
… Well actually I’d just set the car on fire and cut my losses, but you get the idea. 
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Not directly car related, but we had several black widows along one side of the house at the base of the siding. Things we noticed were that they hide during the day, come out after sunset, and know to move away from a light source (like a flashlight) on them and strong vibrations coming at them.
So we hunted them at night, shining lights to the side of them rather than directly at them, and used fast movement using thick gloves that reached them and squished them. Good luck.
Instead of lemon oil, vinegar and coconut oil,why didn’t you just buy raid?
Recent research has shown, Black Widow spiders venom is more potent than previously believed. Researchers also concluded, most bites didn’t fully penetrate the victims skin. If I were you, I’d be a little more careful. We had them in NC when I was stationed at Ft Bragg. Hundreds of them!!
I’m surprised you’d get bit just by driving the car. BW spiders are pretty anti-social, and usually run away whenever I’ve happened upon one. More likely for a bite when reaching under the seat to collect loose change, putting on a sweater, shoes, or gloves that have been left in the car, etc. To get repeatedly bit on the arm? hmmm … maybe they are hiding in the seat belt somewhere.
I was at Bragg for a couple weeks but luckily didn’t see any spiders.
I dunno, I think I would want to see the thing going in the vacuum. I suspect it, or maybe a family by now, is either under the seat, under the dash, or as mentioned in the center console. Can you tell a little from where the web was? I guess I’d be pulling the seats out for a thorough inspection. I don’t know about crawling around looking under the dash but suppose it has to be done-but I’d want to find it.