Where are you getting 220 innings from?
You could easily get him through the playoffs under 200 if you start to manage his starts to about 4 innings for the next couple of weeks.
First of all, you could barely hold him under 200 by limiting him to 4 innings the REST of the regular season if you let him go 4 full starts in the playoffs. Also, they'd never hold him to 4 innings, which would strain the bullpen 18-19 times in the middle of the DC summer, as well as never make him eligible for a win, which doesn't mean much objectively but certainly still means something to major league pitchers.
If you pulled him after 5 innings for the rest of the season, he'd be at 190 or so by the playoffs, then if we made a World Series run would have to throw about 25-30 innings of the highest pressure innings he's ever thrown.
Given his heat aversion, I wonder if the best thing wouldn't be to shut him down now and stretch him out again in September. But they'll never do that for various reasons, not the least of which is that if they left innings for Strasburg in the playoffs and didn't actually make the playoffs, they'd be crucified.