This is actually used in some commercial buildings in places where energy is cheaper at certain times of the day (for example, at night) [2], by freezing ice when energy is cheap, and thawing it when it isn't.
That being said, although this is a super cool project (I have the same inverter, and I made a similar control board ;)), this thermal mass doesn't seem like it would be particularly practical in most cases. Water has a relatively high specific heat capacity of 4.186J/gC, but given the narrow range of temperatures acceptable for a fridge, this doesn't end up being very much - only 79Wh per degree Celsius that the fridge is allowed to swing. If you consider 1C - 6C "acceptable", you only end up storing 395Wh. This is about 30-40% of the capacity of a $100-$200 "deep discharge" lead-acid battery, and is also a much wider range than most consumers would be used to (and may result in frozen veggies, for example).
In order to make this more practical, you really want something that can freeze around fridge temperature. For the same amount of water used above, freezing and thawing the water would store 6,308Wh(!), around 16x as much. If you could get something that freezes at 3C/4C with a similar heat of fusion to water, you could have a much smaller thermal battery that lasts _much_ longer, without the substantial temperature swings you see with your current design.
[1] https://www.eia.gov/tools/faqs/faq.php?id=96&t=3 [2] https://en.wikipedia.org/wiki/Ice_storage_air_conditioning