You should never buy BTC with emergency funds. Many people see the market fall and aggressively use the emergency funds they have at that time, which is completely wrong. Emergency funds are created to deal with unexpected financial disasters. If you invest with your emergency fund at that time, if an unexpected financial disaster comes, then you have no choice but to sell your holdings. In some cases, you can invest with emergency funds.
For example, you have the money to buy aggressively, but you may not be able to invest with it due to some problems. For example, you are not able to transact due to some problems or errors in the bank where you have deposited the money. You know that after some time or within 2 or 3 days, your bank account will be fixed or you will be able to transact again. But you have to be confirmed that you will get the money after some time. In that case, you can invest for some time with emergency funds.
You are correct with your example, that you might have some funds that are more liquid and other funds that are less liquid, and perhaps your emergency funds are the most liquid, but some funds might take a week, a month or maybe even longer to receive them into your accounts, and so in those cases, you might use your emergency funds or anything that might be liquid qhile you are processing other funds that you might consider reserve funds rather than emergency funds, and so if you are using emergency funds for non-emergencies then you likely would be replacing those funds as soon as you receive your other reserve funds and surely there might be various classification of funds that you have, and you presume them to be more liquid than they are so when you go to withdraw from them, you find out that you have a few obstacles , and you hopefully have other funds that are more clearly going to be accessible without obstacles.
To make more emphasis on your respective points and inputs here, i will suggest that if we are investing using an emergency fund, then that fund does not necessarily have to be in fiat, we can as well develop a means of creating a non custodial wallet and have our emergency funds kept on such, because the use of banks could be challenging to an extent that when they placed an embargo on an account, it may not be able to perform any transaction until they are done or settled with why they are doing so.
We can also consider that investing with an emergency fund may come in a quick demand in seizing for an opportunity to invest, just as we do have it form market shocks when it plummet after reacting on a particular cause and we feels its best appropriate that we fall back on our emergency fund in other to take advantage of the opportunity and invest, which i think its another way we can maximize our profit proficiency using another strategy (emergency fund) other than DCA we are used to.