Dragon Beard caramel too soft
The time to reach the given temperature is not particularly important. When making sugar candy like this, the target temperature determines the amount of water remaining in the mixture; as long as it gets to the target temperature, it doesn't matter what it did along the way. (With one exception: If you cooked it to too high a temperature, letting it cool back down won't bring the water back.) 

It's probable that you're seeing the results of a temperature differential. I'm not sure what you mean by "it's a hand-held device, so I can't control precisely whether it touches the bottom of the pot or not" -- can't you just hold it so it doesn't? -- but if you were measuring the bottom of the pot rather than the syrup then the readings will be inaccurately high, and you're likely to get a result that's too soft.

EDIT: So, I should mention one aspect of time/temperature that might matter to some candy-making. If you were to heat a mixture of white sugar and water to 135 degrees extremely quickly, the mixture would be likely to crystallize rather than remaining as a syrup. Heating it slower would allow the sugar time to split into fructose and glucose (known as "invert sugar"), which would not crystallize. In this recipe, though, the lemon juice accelerates the inversion, and in any case the dextrose will prevent crystallization. So while there is a bit of chemistry going on, the precise measurements are not especially important and most of what's happening is just physics. (The temperature is very important.)