Appreciate you're sharing some of these details (thank you). I have been thinking about this same basic idea but with hydrogen and fuel cells but will need to evaluate the merits of electricity -> gasoline vs. electricity -> hydrogen.
Being able to work with existing infrastructure and vehicles and other components designed to work with gas is a huge advantage. And of course the simpler storage of gas vs. hydrogen is nice too.
In terms of efficiency we can do a little back of envelope calcualtions. Prometheus converts electricity to gas at 60 kWh a gallon (33.7 kWh) or about 56% efficient. Hydrogen production via electrolysis comes in at between 44.5 - 55 kWh/kg-H2 (I've seen a wide range of estimates) for a kg of Hydrogen (33 kWh) so about 60-70% efficient.
Of course then there's the reverse process of burning gas or using a fuel cell for electricity production. The Toyota Mirai gets about 68 miles per kg of hydrogen and the roughly equivalent Toyota Camry hybrid gets about 52 mpg.
So boiling this down (if I got this right and my data is good):
Gas method: 60kWh -> 1 gallon of gas -> 52 miles = .86 miles / kWh
Hydrogen method: 55 kWh -> 1 kg of h2 -> 68 miles = 1.23 miles / kWh
Of course this ignores all the challenges to building the hydrogen infrastructure, public fear (warranted or not) about hydrogen, capital costs of both systems, etc. It's also likely the hydrogen conversion gets more efficient than the 55kWh to product 1kg but wanted to keep this conservative.