Why titanium? Very expensive. Obviously copper or steel can't be used. Aluminum may be, but I don't know the effects of it leaching.
I would try to use a plastic in this application. I would think it would be "cool" to wrap 1/2" pvc or smaller flexible tubing inside the perimiter of the freezer. More size options for plastic rather than the titanium. Its not finding the tubing, but it would be finding the fittings and flaring the tubing to fit them. I don't know if a double faring tool used in brake lines would bend it, or if -AN fittings or hard pipe fitting would seal with the titanium.
However, if you are set on titanium, aircraft/ship salvage would be your best bet. Blue Colar Supply in Sac may have something
You may want to research a "Cool can" made for drag racing applications and check out that design.
If that is all the fridge is used for, then you could adjust the temp of the freezer higher or lower based on your need. Simmilar to your heater operation. However, too slow of a feed could freeze the water, and too fast and not cold enough will not have any effect. So carefull conisderation of pipe diameter, freezer setting, and pump needs to be looked at.
I would aim at a low flow, with as high temp in the freezer as possible to cut down on energy consumption. In addition, filling a freezer, say with ice bags, will make it more efficient also. Planned correctly, the setting would be used to keep the ice frozen, and the ice would cool the water as it runs through.
Definitatly an interesting idea and sounds like a fun project.