Flip the script and convey you just could not rationalize $2K a month for rent and you are saving for much more. My first job, after my house purchase was 73 miles away and in hindsight, I wish I slept in my pick-up a few nights a week to lessen the commute. So, you are not homeless but saving money. Get a gym membership and shower there. Eat healthy food at work (when you get a job).
I managed many people and had at least two people sleeping in their offices. They had to be secretive about it, sometimes there would be nighttime office inspections and they would always respond they were analyzing a complex problem, which fit in with what we did at work.
One guy stupidly was walking the hallway in his boxers once, and a federal officer caught him, that took a bit of explaining to get out of that one. My co-workers that decided to live in their offices, also hung out at work all the time, but I think they had a little more flexibility as there were no fees associated with work.
My house caretaker takes care of our property in another state. He lives in the van I gave him. He was homeless for years and slept in a secret place at a local school, complete with an electrical outlet. He went thru great pains to not get detected. After I gave him our old van, he has not been homeless for two years. He takes care of our property (external) in that remote state and has access to our garage for some equipment for his labor jobs. The arrangement works great, and I recharge his Starbucks card in return for his labor. I also have someone I can trust to check on the property and take care of things like fallen trees, flooding, etc. One thing with him, he is “homeless” with van, but since he does not have a 40-hr a week job, he spends a lot of time at Starbucks. That can get really expensive, but it is a tax he needs to pay to have a place to hand out. If my guy wanted to work from my garage, I'd have no problem with that. Perhaps you could create a prop office that looked like a real office from a Zoom/Teams camera view.
Do get a cheap gym membership to stay in shape. It will also keep you spending less money at Starbucks.
Perhaps you can target an eventual job where you can really be dedicated to working a lot and sleeping in your office. This assumes you would get an office. Or a private area. If you could work 4 ten hr days, then you have a long weekend off.
Or, work late and then sleep in your car.
I think you need a full-time job with an office or work area. This will give you a place to hang out and ensure continuity in getting money.
My experience (little) with gig jobs is there are not enough to string together for consistent income to get you into a better place.