Your TS-45 TriStar 45 Amp Charge Controller, takes care of the power coming from your solar panels. It doesn't do anything or work with AC power coming from a pole. It also won't convert the DC from the batteries to AC to power your household appliances. Solar panels produce DC voltage.
An inverter/charger converts DC from the batteries to 120V AC, either pure sine wave or modified sine wave depending on your model. It will also charge your batteries when connected to shore power (120V AC). It does not have anything to do with your solar panels.
A battery charger will only charge the batteries when connected to shore power. It won't change to AC and it won't do anything with the power from the solar panels.
You are asking about doing three separate things.
1 Control DC from solar panels to charge batteries. (control DC)
2 Charge batteries from shore power. (convert AC to DC)
3 Change battery voltage to something your household appliance can use. (convert DC to AC)
As far as I know there isn't one device that will do all of that.
It sounds like you need to do some studying about electrical things in gerenal. Start here.
Look at some of these sites.
The solar controller does the same job as the voltage regulator of the alternator on your engine (only probably better) and no, they aren't interchangeable.