How do you reset the training counter with the console?

There doesn't seem to be one. However, since you're using the console anyway, you could reproduce training with advskill to level up the skill and then remove the proper amount of gold from your character with removeitem.

player.advskill <skill> <nn>
player.removeitem f <price>

This article has the formula for how much gold it costs per level and this article explains advskill in detail.

Well, The answers - including the accepted one, are technically correct, but for the wrong reasons. It's true - you cannot reset the counter; but, you can change how many times you can train per level. Which, unless I'm missing something, would solve your problem just as well.

It's a game setting, by the name of iTrainingNumAllowedPerLevel and obviously, has the value of five. So, with the setgs console command, you can set it to six, sixty or 600 - or any other value you wish.

So, just pop open the console, and type in:

setgs iTrainingNumAllowedPerLevel <value>

Where <value> is how many times per level you want to be able to train. After entering this, the next time you visit a trainer (exit out of training if you have the window open), you'll see the updated limit.