Spares are good. I have 4 packs and 2 extra sets of sticks for our HCH2 fleet. Realistically, I have about 3.5 good packs worth of sticks, and I'm in the process of sorting them.
There is some benefit to stick cycling, but IMHO the reward to effort ratio is off the charts better when working with the whole pack vs. stick. Stick work requires a lot of bench time and coordination to make completion occur at a convenient time to start the next stick. Repeat X20.
With the pack charge and discharge, you get 95% of the same benefit for dramatically less bench time and much less calendar time.
Since you're in no hurry, you could discharge with two very low watt (25-40W) bulbs in series, and you could do multiple cycles. The hardest part will be remembering to check it every day.
I don't have a wiring diagram for the charger, but the second tab in the sheet shows pictures with and without the 12V supply.
Again, I'd make getting your new pack on a 24 hr grid charge the priority. Extended sitting adversely impacts the pack balance, and that's what kills the cells - because it causes the car to run the most imbalanced at the SoC extremes - near 0% and near 100% where cycling can cause rapid deterioration.
Good luck!
Steve