I get what you're thinking. We recently decided to sell a (great condition) '06 Acura TSX which would have made for a fabulous "beater", so to speak. (although I'd have a hard time thinking of it that way)
Logically, it makes a lot of sense to keep a car you know well, that is probably going to be very reliable and inexpensive to maintain, etc. If you were to buy another one, you might (or probably) end up paying more when all is said and done.
But our choice was not solely based on "logic" and since then, we've been sharing one car, so now I have some experience with that.
The Acura was still in its "prime" so to speak as a used car. In other words, the value of the car was still such that it would allow us to reach a significant financial milestone by the time my M3 arrives. That was so motivating, it over-rode the "logic" of keeping the car.
We've been able to share one car without too much hassle, but it definitely decreases one's flexibility and we won't be able to do it for very long. I'm not stingy by any means but I've already had to pay for a day's rental once and another (very expensive) taxi ride another time and I already felt that was a waste. I couldn't imagine paying for a rental x 2-3 months (unless I had no choice).
Still, for us, it was "worth it" to sell the Acura at that time. But I still think a beater is a good idea... so here's my (personal) plan..... (Now that our financial goal is going to be reached several months earlier), I'm going to start saving... for another "beater"! But not another family sedan! A fun little car that I could use as a track toy too.
So... putting that idea out there as another possibility for a "beater" you could use between ED and redelivery.