思ったことをそのまま口にしちゃう - 思ったことをそのまま口にしてしまう - Means "to say what on one's mind"\"to say straight what one thinks".
だけは - I think in your sentence it shows that exactly this 性格(思ったことをそのまま口にしてしまう), is what the speaker doesn't want to be back. I'd say that it makes an emphasis on 性格.
Like as I heard somewhere - 「お前だけは絶対に許さない!」