Did you get your test kit yet? Between that and the geniuses here at TFP, you'd know how many gallons you need to swap out (by testing both the spa water and the fill water). You could also determine how long it would take to fill. So say you needed to swap 150 gallons. That's 30 buckets. You could time how long it takes to fill a 5-gallon bucket from your fill hose. Then you'd know how long it would take to fill 150 gallons from the hose. Could that be done in one day? Say it takes an hour to pull 30 buckets out. Could you fill it back up in time before the temp drops? If there was enough time, and temps cooperate, I'd pull all the water out first, then fill. Avoid mixing old and new that way, and that'll be more accurate than overlapping as you're suggesting.
But if it can't be done in a day, then hmmm. You could try the inversion trick I described. Or you could do X gallons a day (depending on your fill rate). It'll take longer, and use more water, but eventually you'll get to where you need to be.
Come to think of it, are we both over-thinking this?
I'm pretty sure I filled my 12K pool in about 10 hours. I used four hoses. Even if it took 24 hours (I know it was less than that), that'd be 500 gallons an hour. Got three or four hoses you could run into your spa? How long do you think it would take you to pull out 80 5-gallon buckets? Even at one a minute (that's pretty conservative), plus a few rest periods, that's 1.5-2 hours. Fill it back up in an hour (if your fill rate is close to mine). Not seeing the downside to getting this done in an afternoon, and if you empty it all before you fill, you'll be "starting fresh."
And you'll need it filled and running again fast, because you're going to need a good, long soak after pulling 400 gallons of water out of your spa in two hours!! (That's 1.67 tons of water!!)
Seriously, though. Am I missing something? Is it so cold there that having an empty spa, even for a couple hours, would cause it damage? I don't have any experience with that, so maybe someone else needs to chime in here before you attempt any of my hair-brained schemes...
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'nother hair-brained idea. If the problem is draining a hot spa, and refilling with cold water, one of those fill hoses could be run from your water heater, mixing with one or more cold water fill hoses. That'd be warmish water for the fill...