Since these batteries would be rather expensive, I imagine that you would have to pay a refundable deposit to the tow-a-battery network (Shell, Esso, etc.) that covers the full cost of the tow-a-battery in order to rent one.
The biggest down-side is that a rear-end crash could be rather dangerous. I don't see a way around that without making the tow-a-battery a vehicle in itself.
Edit-here’s an article about a company doing this
https://electrek.co/2020/02/19/french-startup-proposes-batte...
as an aside, the cybertruck is (probably) pretty bad at towing. see this video for an interesting argument: https://www.youtube.com/watch?v=S4W-P5aCWJs
A car cruising at highway speed is lconsuming something on order of 10x as much power. Given how much smaller cars are than apartments, it's an even greater difference.
So to answer your question: not much. Heating in cold weather is much worse on range, as it's not just the cabin but also the battery that needs to be maintained at an adequate operational temperature.
However, in practice I think this would be equivilent to towing a small tanker trailer full of chemicals or gasoline. It would require plaques indicating the contents, special insurance, and all kinds of BS at the RMV. Imaging if one decoupled on the highway and drove into oncoming traffic at 60mph? It would essentially be a rolling bomb. This happens quite often with camper trailers.
I think engineers would need to emphasize redundancy, failsafe measures, and automated attachment/detachment capability that does not require human interaction to safely accomplish. Then you will convince CA that it doesn't need a ream of paperwork to register and then the rest of the country will adopt it.
Personally I see this as completely new tech from the ICE. That means it really ought to open up new usage and user experiences than the outgoing tech. And those changes should be perceived as better in some way to really take off.
Edit-forgot the link: https://electrek.co/2020/05/06/mobile-ev-chargers-and-vans-g...
I don’t know how running the AC impacts range, I would like to know. Places with extreme weather and major population centers can make an otherwise short drive really taxing on resources through ac use or just being stuck in place for extended periods of time. Like how peak LV traffic can turn a 15 min drive into a 50 minute drive in 120f+ heat.
Tow-able batteries would make wide use cases like apartments more viable too.
Edit-typo