If you're looking for reliability, then battery should be the one place you spend some money, as it is the heart of the system and has to supply all the power for every job you need the bike to do for you, reliably, 100% of the time for as long as you need it to.
You can use the motor and trip simulators at ebikes.ca to determine the power level you will need to do the climb, and estimate teh wh/mile it will take so you can determine minimum battery capacity required.
You don't mention your weight and that of the bike and anything else you'll be carrying, so I couldn't do any simulations for you. Power will increase based on weight, speed (which includes winds, because the air resistance is what counts), rolling resistance, and slope.
My guess is that for the worst 12% grades at say, 20mph, with a medium-light rider for something like a couple hundred pounds total bike/rider/system weight, it will take at least a couple thousand watts of battery power, maybe 3/4 of that at the motor (wasting the rest as heat in the system), and at least 100wh/mile battery usage. That's assuming good pavement and road tires inflated correctly.
If you had a 52v (14s) battery, it would need to be able to sustain at least 40A continuously to get the couple thousand watts for the system, and at 100wh/mile it'd take at least 1kWh to go 10 miles (of course, you won't use nearly that much capacity if the 12% slope portion(s) aren't much of the total distance).
But it still requires the system be able to provide the max power continuously, even if it's only needed for a short run...so the battery has to be able to do that with minimal voltage sag (the more.sag, the less total power available).
For the dirt or poor paving, etc., or offroad tires or low tire pressure, etc., it will take more power, and larger battery capacity, to do the job.
Same for higher weight, faster speed, non-tailwinds, etc.