Just because they're faster, doesn't mean it will be a cheaper charge. Your Clarity battery will take a certain amount of kWh to fully charge, and whether they get there fast or slow, the $/kWh is not going to change. If you're on city water, is it cheaper if it comes out faster? The only thing affecting charging cost is the rate and if your utility offers cheaper rates at certain times of the day. If you go with a Level 2 charger, the costs for purchase of equipment and installation (which could be $1000s) should be added to your Clarity costs and amortized over time.
For most people level 2 will likely be cheaper for several reasons, but it certainly depends on the individual situation whether the savings will ever pay back the cost of upgrading to level 2 which includes the cost of the EVSE and the cost of installing a 240V outlet. Your example of $1,000 for the upgrade is certainly possible but for many people it seems to be closer to half that, especially if any rebates are available, and when they can get a 240V outlet installed for a reasonable price and not get gouged. I have heard it's better to not tell the installer that it's for an electric car because inexplicably the quote goes much higher. People who just say I need a 240V outlet installed and don't say what it will be used for seem to get lower quotes. Go figure. Of course installation cost also depends on the length of the run from the panel to the new outlet location, accessibility for running the wires, whether the panel needs to be upgraded, etc. all of which can certainly drive costs higher, but for many people the upgrade is relatively easy.
Also with the Clarity you only need to buy an inexpensive (approx $200) 16 amp level 2 EVSE which will do a full charge in about four hours. Clarity will charge in about two hours at 32 amp however the additional cost for the higher power EVSE is not really worth it for most people since they would rarely benefit from the additional charging speed of 32 amp over 16 amp.
As for the savings, the water example doesn't really apply because charging at 240V is more efficient than 120V, but from what I have heard the difference is somewhat small so by itself it would likely not be enough to ever pay back the expense of upgrading to level 2. But it would help a little bit.
Also many people drive enough miles each day to fully use their EV range and then some, and since for most people depending on the state they live in electric miles are cheaper than gas miles, the more they can charge the more money they save. The Clarity OEM cable at 120V takes about thirteen hours for a full charge, and sometimes you don't always have that much time, like if you get home late and have to leave early the next morning. Or someone who typically arrives home from work with 0 EV range but will be driving somewhere that evening so they can only charge for an hour or two, that might not be enough time to put in all the electricity they need for that evening's driving. Again this is very individual both on driving patterns and also local cost for electricity and gasoline, but for many people the savings can add up if they can do more charging with level 2 than they could with level 1.
The bigger savings for the OP is because they have a TOU (time of use plan) with a much lower rate in the middle of the night. I have a similar plan where I get half off the normal rate, but the hours are limited from 11:00 pm to 7:00 am. When my battery is near empty that is not enough time to fully charge, which requires an additional five hours at a rate that is twice as high as my overnight rate. Although I don't need to fully charge most days so I am not as affected by this as others would be who do a lot of EV miles each day.
For many people, even after factoring all of this in it likely will not fully pay back the cost of upgrading to Level 2. But there is also a convenience factor to be considered. Preconditioning for example works much better with level 2 than with level 1. With level 1 you cannot run climate and charge at the same time like you can with level 2. Also most owners simply prefer driving EV over HV, and if level 2 enables more EV driving, then that is worth it to them as it helps them enjoy driving their Clarity more, even if there is not an overall net savings. However for others, depending on their individual situation, they may decide that it's not worth it to upgrade to level 2.
That's one of the huge advantages of PHEV over EV that is not always mentioned, that Level 2 is optional for a PHEV, whereas for many people who have an EV level 2 is required if they want to avoid using external charging stations. The question that I am asked most often about my Clarity is if I can plug it into a regular wall outlet. I can simply answer yes, whereas an EV owner will likely have to give a longer answer to that question, explaining that yes they can but that for their situation they needed something faster at home and so they installed a 240V outlet.