Both are possible, it is just that since we have no way to control the events that happen around us, being too aggressive in our investment strategy can be very detrimental, if we find ourselves facing a difficult set of circumstances.
So the safest path is to build your emergency fund and only then begin to invest in bitcoin, however such an approach has an opportunity cost and you may miss buying bitcoin at an attractive price, so I can understand why you may prefer to take a hybrid approach.
I prefer to buy Bitcoin at an attractive price, namely when the price has fallen or is still in a fairly low area historically for several months. However, the investment approach as you say is also very good because everyone is required to have an emergency fund even though the fund is not used often every day. But I actually prefer to buy Bitcoin first, even in a smaller amount, rather than preparing a larger emergency fund because I like holding Bitcoin even if the amount is not that much. Because preparing an emergency fund also takes time and it could make us miss the best price on Bitcoin so that we don't have time to buy at a more attractive price.
Maybe we have certain criteria's about saying about attractive price since maybe you mean upon buying when the dip comes. But for me I guess this is not good since we don't know about on what will be the lower point of that certain scenarios. That's why I'd rather to buy whatever price it is as long as I have funds to use since I'm not aiming for long term on my bitcoin investment and for sure any figures will do since there will be a huge changes will happen on the price of bitcoin on 4 - 10 years time span and with that we have good chance to earn more.
We don't need to put up huge money instantly on emergency funds. But rather you can put some money you can afford to hide and if you are consistent for separating some funds for this purposes then for sure that the amount you hide would provably became bigger but set a target that that fund will last up to 3 to 6 months you will be fine when there's emergency situation came.