Why trust someone else if you want it to be anonymous? Buy a proper GPU, run a local model and that's it handled. Basically all open source inference engines support OpenAI's API format and act as drop in replacements.
How much do you think those API calls will cost you that require someone else to buy the GPU, maintain the server architecture, and also take a cut in the process? It only takes a 3-4 months to amortize if you want the instance to be up 24/7. And in the end you can still resell the GPU.
I rent spot instances on GCP with big GPUs on demand. It works quite well and since I'm just playing around I pay a very small amount per hour and kill it after I'm done.
Does this work on-demand? Could I build something I can query from my phone as needed?
Depends on your budget. If it's the absolute minimum, it won't be real time responses. As provisioning infra isn't instantaneous.
Just want to point out that $1000 is nothing to a lot of potential bad actors. $1 million is nothing.