I've only done some research, but if you have the space, you can skip the soda stream and carbonate in a corny keg in large batches. The lid of the keg has an input for co2 and an outlet for liquid. You should probably have enough space in the fridge for the large keg. There are smaller keg sizes but I'm not familiar enough to name them.
You can also find continuous carbonation lids but there are constraints on the water pressure being higher than the co2 pressure. My well water is 30 to 40 and the recommended co2 pressure is about 60.
On the whip (cylinder to cylinder filling tool) you linked, you can only ever equalize the pressure between the two cylinders. Most gas cylinders I've seen start around 2000 or 2200 psi if they were professionally filled (as in by a gas supplier like airgas). If your home brew store used a whip to fill your cylinder, it's going to be below their starting cylinder pressure. Each fill is going to lower the source pressure and generate a lower pressure in the filled cylinder. Vacuuming the cylinder should get more CO2 out of the source, but won't fill the small cylinder more. If I understand correctly, because you're pulling the remaining co2 out, you'll get lower pressure in both cylinders at the end because more gas has to leave the source to equalize the pressure.
Tbh, I'm more scared of the whip than a regulator and a hose after it. If you don't have a regulator on your source cylinder, whatever comes next has to handle the full cylinder pressure. In your case, all parts of the whip are at cylinder pressure. Regulators can handle full tank pressure and make it a safe level for the hoses so that in the event of hose failure, you're only leaking 60 psi (or whatever the regulator is set to).
I used to work with tanked oxygen for torch work (not professionally), but I think the above is correct. If not, please correct me.