If you only plan on his taking it twice, I would go with August. Gives him all of summer to prep on a more measured basis. IMO, the best prep is taking full practice tests under real conditions and then immediately review/understand why the mistakes. Taking tests relatively close with no additional prep is unlikely to result in much gain, unless the first time was low because of nervousness and unfamiliarity with the test.
If you would consider a third test, then go with June. If he hits his target, then he is done. If he is still short, he can prep for August.