So just tried stoking the first time, and noticed the cost jumped for the next stoke (1k for the first, 2k for the second). Wanted to know - does this ever reset? Or does the cost just keep getting higher and higher as you use the feature?

It resets daily. The most expensive it gets is 5K and it stays that way until LW resets the next day. Then it's 1k again and then back to 2, 3, 4, 5.

One way to try and avoid the consumption of available stokes by one-ore shards is to wait to execute the stoke until after smelting. As an example, go to the dig site before stoking, then when your wagon is full and lets say you have 4 of two precious ores, 5 of a lesser precious ore and two moonstone shards, go then to the furnace without stoking. On smelting completion you will have the lesser precious ore bar and two moonstones waiting whereupon you can stoke and it will begin smelting again giving you the two precious ore bars while reserving 4 stokes for the next mining attempt. It doesn't always achieve maximum usability but can increase the odds over going in stoked ahead and having most consumed by one-ore shards which happen more frequently the deeper you mine.