The first thing you need to determine is how hard your water actually is. A professional lab analysis costs about $100 and would be cost prohibitive and impractical if you're moving around a lot. these tests will show all impurities in your water in addition to overall hardness. If you're just concerned about hardness, I would recommend one of these:
I have had my home water lab tested, and as far as the hardness level, found this test to be very accurate, and they will do many tests. Don't waste your money on the test strips.
At the low end of the hardness scale is 1.5 grains/gallon. The regeneration frequency on a salt type softener is based on total hardness of the water and the capacity of the softener measured in grains. Some of the RV type I've looked at have a 16,000 grain capacity which is excellent. The modern residential models calculate the regen based on gallons used factored with the grains per gallon that you enter in the controller, they're based on demand rather than time. The once per month the manufacturer recommends is likely an educated guess based on average water hardness and estimated usage.
If you want to be accurate on when regen is required, install a small meter in the water line ahead of the softener, and if you know the hardness level of the water and capacity of the softener, there are online calculators to determine frequency recommendations:
flixwater.com
You could also just run another test to determine if your hardness level is creeping up. If you find a regen is needed during freezing weather, just wait until it warms up to do it. Your water won't cut off, you'll just be using hard water for a day or so.
The salt free models have some drawbacks mostly mentioned before on this thread, additionally they just don't work past a certain hardness level. I'm not sure if the filters you have ordered are this type, but I tried a "limeblaster" filter on my steam humidifier and was pretty disappointed with it's performance, but my water is extremely hard; 15 grains/gallon.
In addition to accurately calculating your regen cycles, you could pipe your system so that you're only softening the cold water entering the water heater. This would protect the water heater and provide soft water for showering, dish washing, laundry, etc. Who cares if you flush the toilet with hard water or the small amount of cold mixed into your shower is hard? It would also further extend the regen frequency. If you want to treat your drinking/cooking water and ice, install a small RO unit under the sink.
I highly recommend calling USWater Systems and talking about your issues. Their customer service is awesome and their expert advice is free, you don't have to buy from them. It would be a good idea to know your hardness level prior to the call.
You very likely can't soften your water effectively without having the tank in your basement. If your major concerns are the regeneration, you can take steps to control that to a great degree.