why not just empty everything into your bitcoin investment at once since it's already looking like the emergency funds can be invested.
No emergency fund shouldn't be invested in bitcoin, because emergency fund has it own purpose and role to play when the needs arises, emergency fund are to be use only for emergencies and not to be invested in bitcoin, investing your emergency fund can endanger your bitcoin investment, if emergencies arise and there is no emergency fund to handle that emergency It may lead to selling off your bitcoin investment at early stage of accumulation which might likely land you in lost. Only invest in bitcoin with your discretionary income and any other money aside your discretionary income shouldn't be invested for they have there own purpose and there role to play when the need arises.
I cannot agree with you, if you discuss like this then the image of Bitcoin investment discussion may be ruined. Emergency fund will not be used only for our own needs, it should be used in any situation. You notice that you have participated in Bitcoin investment and have an emergency fund, and you have fallen ill, then at the moment you have sufficient emergency fund.
Which you can use for the next few weeks, and you can withdraw money from the emergency fund for recovery and use it for medical treatment and continue investing Bitcoin in the regular DCA method. Then your treatment and regular procedure are basically to be followed, the emergency fund is deposited to recover from all these dangerous activities.