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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.
I would not support anyone using their emergency funds for non-emergency purposes. I clearly understand your pattern of slight flexibility with respect to maintaining consistency in your purchases bearing in mind that those funds are present and in processing, but we know that these processes aren't totally simultaneous with our timing. Now let's consider a scenario that the investor has already invested a large part of emergency funds into his portfolio and there was an emergency within that time and the funds in processing remained stuck for a while, you'll agree with me that his portfolio is at risk of being tampered.
I would rather suggest a different path, that he waits and keeps track the amounts he should've invested into bitcoin and as soon as those funds are processed, the investor should be disciplined enough to compensate for the week(s) he missed out on with his accumulation amount. Also it is always very good to keep these funds for investing in stable coins as soon as they are available rather than banks, so that purchasing bitcoin can be easier.