allows me to trade immediately on money it knows I am getting, so you should be able to buy the shares to cover using the strike price they know has to be coming in order for you to have to send those shares.

"Should" is doing a lot of heavy lifting here.

Yes, you can do that with a sell and a buy in the same account. At most brokers. But not all. I had one broker that would not allow you to buy until the sell has settled.

But, they won't let you, say, transfer the money "it knows I am getting" before settlement to another account at the same broker and buy in that other account.
Under Balances, they show "Available to trade" and "Available to withdraw". The first is the unsettled proceeds from a sale that they know you are getting but have not gotten yet.

Anyway.......it comes down to the timing of when they credit your account with the proceeds from the exercise, in relation to when your shares are delivered.

A quick search show "Expiration time occurs on the third Saturday of the expiration month at 11:59 a.m. EST."
You could not buy the replacement stock to deliver until Monday morning.
I couldn't find anything that directly said when the money hits your account.
You will have to depend on your broker to credit the money as "Available to trade" on Monday. But the new shares are not in your account until settlement on Tuesday, whereas the existing shares leave your account on either Saturday or Monday, or Friday at the close (see below).

Interesting question. I think you would have to try it to see exactly what your broker does. For sure you will NOT be able to get this question answered by the telephone answerers at Customer Service -- too technical.
I am skeptical that you could buy shares after expiration and say that those are the shares that you delivered instead of the shares that you had owned. I think you would only be on solid ground with shares that you bought BEFORE expiration.

Which brings up another interesting question: Can you deliver shares that you bought but which have not settled? Techically you don't own the shares until T+1 settlement.
Which your broker will nastily inform you and slap your hand if you buy a stock and sell it the same or the next day in a non-margin account.

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I found a Q&A that said "I've never come across a broker that allows you to specify lots when you write covered calls but I've only used about half a dozen brokers for options in 35+ years. And if that ability exists, the key question is, does Vanguard allow you to do so?"
"The only way to find out is to call your broker and ask if you have the ability to designate which of your shares will be assigned. I doubt that you can."

When my Mom's full service broker did CC's for her, at exercise they coded it as a sale of 100 shares at the Friday close. Obviously that was the old shares, not new shares that you couldn't buy until Monday.
Of course, being a full service broker, they charged a full commission on the "sale". I complained to them and was told "That's how it is done, the stock is a sale in the CC account and a buy in the call owner's account."