How to use 整 as in exactly
整 should be used after the measure word, so 三百八整人 is not correct.

However, 三百八人整 is weird too. 整 is usually used at the end of the phrase to prevent from adding new numbers to change the whole meaning, especially in financial affairs. Such as

三元 -> you can change it to 三元五角八分
三元整 -> you can't change it to 三元整五角八分, which is wrong expression.

Originally 三百八人 can't be changed to 三百八人零点五人 and so on, so need not to add 整 after that.

BTW1: It also can be written as 正 in financial affairs.
BTW2: As @user58955 said, 三百零八人 is more natural than 三百八人.