Yes not having a stable source of income shouldn't impede us from investing in bitcoin as long as the needed discretionary income is available. The most important thing is if the needed discretionary income is there. Just like you said having a stable source of income doesn't mean that the needed discretionary income will be available. Therefore the most important thing is to know if we have the discretionary income to invest with ,as for having a stable source of income we can work towards it . We can stabilise our source of income while investing in bitcoin .
To invest, our source of income does not have to be stable. A person can have any source of income, if he can find a discretionary one through proper financial management, then he can start investing.
Let me give you a real example, there was a person in our area who was a daily wage earner. He used to stand at a place and if someone liked him, someone would call him for work. You cannot call this work of his as a stable income. He can invest even if he wants to, he can invest with the amount of money left after paying all the expenses from his main income. A stable income is not something that is needed for investing. A discretionary income is needed for investing.