Current wiring is: Line 1 and 2 are the panel and pump; Load 1 and 2 are the SWG
I am not quite understanding what your system is actually doing.
Line 1+2 should be the incoming power, not switched by the timer/freeze protection, and Load 1+2 are switched on/off by the timer/freeze protection.
Your description above sounds like pump and panel are on permanent power, and the SWG is switched on/off by timer or freeze protection events. That seems to imply that your pump is either permanently running, or there is another feature in your system that turns the pump on when the SWG turns on.
I don't think that this setup is achieving your goal as described here:
We need the pump to come on independent of the SWG so we can run in the winter.
And just for my understanding, which of your equipment is running on 120V and which are on 240V (ignorant Aussie here, we just run everything on 240V, so I am not familiar with spotting which devices typically run on which voltage)?
In your last picture there are two red cables on Load 1, but only one black cable on Load 2. Could it be that both, pump and SWG, are connected to Load? One of them is on 240V and needs to be connected to Load 1, Load 2 and Neutral, and the other one is on 120V and only needs to be connected to Load 1 and Neutral. And only the panel is connected to permanent power supply on Line 1 and Neutral (if on 120V) or Line 1 Line 2 and Neutral (if on 240V).
That would mean that both, pump and SWG, are turned on by freeze protection, but the SWG doesn't produce chlorine due to cold water shut off.
Looking through through the manual
...makes me think that the pf1103t should be capable of copying your setup.
But this statement in the manual worries me a bit:
Both models are designed to operate either on 120 Volt or 240 Volt, 60 cycle. When used on 240 Volt however, the neutral wire must be installed and it is not recommended for controlling 240 Volt GFCI protected equipment. For such application, use Model PF1102T Time Control with Freeze Protection.
I assume that all of your equipment is GFCI protected? I am not familiar enough with US 120/240V setups to understand the background of the GFCI caveat in the manual.
I also don't understand US wiring well enough to see why the GND in the pf1102t couldn't be used as Neutral.
Maybe someone like
@JamesW would be the right person to help here?