Two thoughts on the subject:
Thought #1 - Let kinch use up all of his draft picks, but he will need to drop 2 players at the end of the draft, or when he drafts his 19th player he then has to drop someone immediately (basically as he drafts the 19th player) and the player he drops can go back into the draft.
Thought #2 - After kinch drafts his 18th player he then loses his last 2 picks. This still allows kinch to trade picks to move up, but would prevent him from going over the mandatory 18 man rosters.
Of course this would apply to anyone who could end up drafting more than 18 players.
And a reminder regarding the other side of the trade, ie Niagara, those who don't have an 18 man roster at the end of the draft can then pick up players AFTER the draft to fill their roster. This will be during the free add/drop period between the end of the draft til a few days before the season starts.