Investing with your emergency fund is not a total bad idea at all because after you have might paid all your bills and the money remaining becomes your discretionary income and inside that your discretionary income you can still generate out your emergency fund and started investing in the last 6 months and think you to invest using your emergency fund so that it multiple more income since you are not using it at the main time, you can invest with the emergency fund and tage it emergency fund which means it can be pull out if incase of unforeseen circumstances that might happen, so the reason why we should invest emergency fund is to back your really investment incase of anything, so you can decide on what to do with your emergency fund as per dividual convenient or best know to you, if actually you needed your emergency to in cash but left with me, i will say that once you progress with your Bitcoin portfolio that there are always a need to invest to have emergency fund.
I find it difficult to understand everything that you said in your post because you are mixing up your words and giving the wrong use of an emergency funds. Emergency funds should only be tampered with when there's a real life emergency and not to use to buy bitcoin. Your discretionary income is what you should use to invest into bitcoin continuously and persistently overtime.
What if you use your emergency funds to buy bitcoin and the next day, you are hit with real life emergency, you will have no option to sell your bitcoin at loss, if the price dips below your entry point. Any investor that uses his emergency funds to buy bitcoin and neglect the importance of emergency funds is only gambling and not investing because you will definitely sell your bitcoin when you are hit with real life emergency.
You cannot use your emergency funds for whatever thing that you like because it's not flexible. Your reserve funds is what you can threat like that because it's flexible. It's also wrong for you to believe that a real life emergency wouldn't occur because you don't know what will happen in the next minutes which is the reason why an emergency funds is very important to be on ground at all time provided that you are still accumulating bitcoin.
It takes up to a year and more to build an emergency funds of three months of your expenses, if you exhaust your emergency funds how will you cope with your bitcoin investment. Don't involve yourself in a practice that will make you start reducing the size of your bitcoin portfolio instead of increasing it because that's not the goal.