The "charge only" button will turn the charger on and enable AC passthru when you are plugged into shore power.
"Auto" will essentially turn the inverter on, how it functions will be a product of if you have shore power plugged on or not. If shore power is not plugged in it will enable the inverter to provide AC power. when this switch is left on it will at a minimum be checking if AC power is needed, if the threshold is met it will start inverting. The load that it turns on is controlled by a load sense dial. This can save some power, but the better goal is to only enable the inverter when wanting to use an AC appliance. SMB has a habit of sometimes not mounting the remote panel "APSRM4" in a convenient place.
If you are plugged into shore power with auto on, the AC will be passed thru and the inverter will be in stand-by mode. If AC power drops from the shore power side the inverter will pick up and provide AC power to your connected equipment.
The inverter itself has a 3 position switch "charge only, off , auto" , for the remote to work this switch needs to be in "auto" and the remote plugged in of course.
Your wiring looks even worse then standard SMB wiring. I would check to see if you can find the model # of the tripp-lite. I am not sure if it's the angle of the picture, that makes it look a little smaller than I would expect. I also would have expected the cable to be 2/0 gauge for a 2000 watt inverter. So just more to check.
What is the max current values can the hall sensor can manage, will it support the heavy loads when the inverter is being used? With the hall you can go on either positive of negative, just need to get the direction of the sensor correct. You will want to make sure that it is placed right before the cable goes thru the floor to connect to the battery. Also make sure their are no other power using cables connected directly to the battery. I would budget for a proper shunt based SOC meter in the future.
As far as moving from the sure power unit, the BlueSea 7610 is a great little unit. If you have the upgraded alternator, or think you will be putting on larger alternator in the future move right to the 7620 or 7622. The 7610 is for alternators up to 120 amps, to meet this current rating you would need 1 awg versus the 2 awg you have. The 2 awg would derate to 100 amp. This most likely would still work great for you, but because the 7610 is FET based they require larger size wire in their installation documentation. It is a heat transfer thing. Obviously boywonder has used one for years.
The 762X units are magnetic latch relays so they wouldn't suffer from that. Earlier in this document I lined a pdf describing upgrade from Surepower to 762X. The terminals are larger on the 762X units so you would need to replace those ends. Make sure you put a fuse on the ground wire to the 762X. If you chose to use start isolate, then the existing start isolate wire for the SP 1315 would attach to that input not the red control wire. I highly recommend the switch to be installed, but it will work with out.
As boywonder said a properly rated SPDT switch would work,
or use a SPDT relay with a switch to control relay.