The simple answer to the difficulties raised in this question is that establishing a startup is a full time endeavor. By definition one cannot work full time employed for a third party and full time in their startup.
If you are accepted into a formal accelerator program, you will not be able to leverage the benefits of the program unless you are available full time. Similarly, significant investment is unlikely from investors if the key founder is not working full time on the opportunity. Additionally, concept and seed level investors generally desire that the startup use investment proceeds to grow the endeavor, not pay salaries, i.e. your earning your sweat equity. That said, most are comfortable with subsistence salaries once the startup is demonstrating some traction. Note however that most investors will avoid startups with founders demanding six figure salaries unless they have significant sales and scalability which is proven.
We would suggest working part time on the startup while you save enough money to facilitate leaving your full time job and moving full time to the startup. Once that occurs, you will be able to focus on growing and accelerating the startup, regardless of process chosen.
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