The new battery (700 mAh) comes up with 3.7 * 0.7 = 2.6 Wh, so it should be capable to power your device.
Your original battery (1.41 Wh) comes down to 1.41/3.7 = 380 mAh, or it is smaller than CR123A. It should be expected that your built-in charger uses no more than 200 - 380 mA charging current, which should be fine for CR123, it will just take a longer time to re-charge.
More challenging would be the handling of "white" wire. I assume it is a plain thermistor, although it could be more complicated. You can measure the resistance between thw white wire to ground (black wire). If it comes up as, say, 10k +- 25%, then it is a thermistor. You would need to fool the white wire with the same value, plus-minus. Without the termistor the charger would think that the battery is overheated (or broken), and would refuse to charge it. The negative side is that the thermistor was there for a reason of extra protection, and your new setup will be lacking it.
However, the Nest thermostat is a fairly popular product, so the replacement batteries should be freely available. It is advisable to get a direct replacement.