Months later...
It looks like what I want to do is create a box file, so checking out the wiki, I modified the instructions to create this command, which seems to do what I want:
tesseract text_image_file output_file_name makebox
Output looks like this:
C 261 2453 285 2480 0
A 287 2454 312 2480 0
P 315 2454 334 2479 0
I 337 2454 347 2480 0
T 349 2454 372 2481 0
O 374 2454 402 2480 0
L 406 2454 426 2480 0
I 429 2454 439 2480 0
N 442 2454 471 2480 0
E 473 2454 494 2480 0
So now I need to process this output to get the letter heights (element 4 - element 2 in each line) and then grab the median.