Three pages in and only one person replied to your original question- "can you just hook up the relay directly to the AC line voltage"?
Yes, you can. I think you must've misread the datasheet for your relay, because 75 ohms is WAY too low for a 120V relay. That's a reasonable value for a 12V relay, but not 120V. Check out page 3 of this datasheet:
https://cdn.automationdirect.com/sta...s/78relays.pdf
I just picked the first one in the list. Their 120V coil relay has a resistance of 4.43 kOhms, meaning it consumes under 2 watts of power. That's going to be fine to be left on all the time. If you add a power resistor, you'll reduce the voltage across the coil and it won't work.
Also, I doubt you can measure the resistance of the coil with a normal multimeter. Maybe you can, but I suspect the "resistance" they're quoting is the AC impedance at 60 Hz. (Basically, the AC version of resistance is impedance, so the math still works with Ohm's Law). If you measure an AC relay coil using a normal multimeter, you'll get the actual resistance of current to flow to DC voltage, which isn't what you want.
I'd do your original plan- hook up the relay to mains power, then if mains drops out it connects in the battery/inverter solution. Make sure to test it a couple times per year to make sure the battery is still working. This is a better solution (IMHO) than letting the battery run it full time.
If your house power dies, you will know it immediately. If the battery dies and you don't have an extra alarm, you won't know it until your fridge stops working.
Basically, the "critical failure" point here is for the fridge to lose power. With your relay solution, you reach critical failure with *two* things going wrong- your house power dies, AND your battery dies. With the "let the battery run it all the time" solution, you reach critical failure with just *one* thing going wrong- the battery failing.
Now, in any given case you could argue that having option two means that you can detect the battery failure *while you still have house power* and can therefore fix it. With the relay solution, there is potential for you to not notice the battery failure until both items fail.
You can only plan for so many contingencies, but I think I'd still go with the relay solution and put an alarm on the battery itself. Then you're not worrying about the inverter and battery providing fridge loads 24/7, and will notice when the battery dies.
Either solution will work- you can just decide which failure scenario seems less risky from your point of view.