I still use a very similar system to what you have on hand, in fact i started out with what you have on hand, except the 12v pump was about 5 gal at 45lbs
Just siphon down to the pump, the pump
should have
enough of a draw to start the siphon, but i wouldn't recommend the pump be higher than the water barrels.
Send your water wherever you want it to go.
You can by bass the existing pressure switch just by shorting out the terminals on the switch- a small jumper, or just pull both wires from the switch and connect them together, then install your other pressure switch somewhere in the water line on the outlet of the pump, so the wiring goes one pump wire to battery, one pump wire to new switch, one wire from other terminal of switch to other pole of battery.
I do not recommend running at 115lbs--most plumbing is 60 lbs max. You may be able to get a higher flow using the lower pressure setting, but don't count on that. Better to just buy a new pump that is the right pressure setting. Are you still trying to make RO water? Or are you just inheriting these parts from an old ro system.
I have had quite good success without a pressure tank, but it does cut down on the cycling of the pump, that tank just hooks into the line as a splice between the switch and the rest of the plumbing.
I'm still using basically that same system, rain water feeding a cistern, siphoned to a 50 lbs pump, then sent everywhere, hose outlets, bathroom, kitchen, gardens, the only shortcoming is with more people comes greater demand, and at some point you might want a bigger pump, or separate systems
Filters and freezing considerations need to be addressed as needed, diaphragms don't like leaves etc, pipes don't like to freeze.