It must exist!
Or rather a close approximation. You may have to compromise somewhere but perhaps not...
In contrast to the above I kinda feel that a pop up/one piece fold double would suit you well as they tend to be lighter to push than tandems and definitely lighter up kerbs. There are carrycot options on some of them, Baby Joggers being the obvious brand. Tandems lighter to lift into the car purely by virtue of being in pieces when you put them in really.
The very lightest to push doubles often seem to lack a carrycot option eg I would suggest a Nipper 360 double (the newest narrowest one, 72cm should get you in most places) and meet all of your other criteria except the carrycot bit.
Slightly wider and heavier options but with carrycot available include Baby Jogger City Mini Double and City Mini GT double. I had a normal BJCMD a few years ago and found it easy to push and handle, I think the GT is heavier to lift etc. but possibly better to push due to the air wheels? The GT is over budget, the normal one about on the nose or under.
The narrowest side by side available, the mountain buggy swift, is over budget and quite a lot over with a carrycot. It is fairly easy to push but not a dream and surprisingly large and heavy folded for a narrow double. I would struggle to put this in and out of the car regularly (and I am far from um, little!)
There are some good tandems but the better ones tend to be expensive. The Smyths Toys Dimples Duo is good (there is a pic of one on my profile, toddler has loads of vision, baby lies flat facing you though not in an actual carrycot) but I think you would struggle to lift it into the car and also tip it up curbs. I had no problem with curbs but my mum did, she is 6 inches or so shorter than me and it definitely had an effect. I don't know if this would apply to a lot of tandems, they are inherently that bit more of a struggle to push and handle at times. I have a Jane Twone atm, which actually does meet your criteria really, I do love mine but the seats are very close when you get to the seat/seat set up as oppose to seat/carrycot. I have cheated by using a 2nd main seat, which totally transforms it but is of course completely not recommended! I think a BJ City Select kind of meets your criteria but it is heavy to push and very very hard to lift up curbs. I have heard mixed things about various phil and teds, generally speaking that the older models were better but haven't directly experienced one so struggle to comment.
I'm still thinking, I'm sure there are more but also keeping in mind you need it soon, which rules certain things out. I've sent Tiggy in this direction, she is the true oracle of Doubles knowledge 