My son did HF Session 1 in CA (1 inning on day 1 and 2 innings on day 2 -- perhaps 25 pitches on day 1 and 50 on day 2) and was scheduled to attend Showball in NY. I had the same concern so I called Showball and worked out with them the following schedule -- no bullpen on day 1, and on day 2 they agreed to have him pitch the last two innings of the first game, and the first two innings of the second game. So, essentially, 4 innings in a row.
With any of the camps, you are the paying customer, your son should do as much (or as little) as you feel comfortable with. Another example: my son skipped the medicine ball throw, shuttle runs, and vertical jump testing for pitchers on day 1 at HF. IMO, that testing wasn't likely to improve the likelihood that he would pitch his best in the late afternoon game, so he skipped it.