Are you sure you don't mean 'remorsefully'? Dictionary.com says "No results found for 'remorsely'."Um... That is a word. You may be more accustomed to seeing it's cousin 'remorselessly' but lack of familiarity does not make a word incorrect.
Instead of 'in a manner lacking remorse', which is what 'remorselessly' means, 'remorsely' means 'in a manner of remorse' or 'in a manner of self-reproach, pity, or compassion', to further define it.
In other words, despite the fact that I am logical, I sometimes feel guilt, compassion, or pity when I'm using that logic against another. I am remorsely logical. Others have told me so and I adopted the phrase as my user title because it was not only apt but also because I liked it.