Publishing: Is working at home perceived as unprofessional?
No one will know that you are unaffiliated since you are free to put your research institute as an affiliation. Your institute might even be upset if you don't since you did this work while in their employment even if your contract doesn't give them rights to it if done on your own time at home using equipment that does not belong to them. If you put them as your affiliation, someone will eventually notice and may come asking questions. As such, it would probably be best to show this work to your employer through your current line manager/PhD supervisor and ask for their advice. It's likely that with some help you and your advisor will be able to publish this article together. This is how it works in the "real" world. 

My favorite publication of my own was mostly done at home and in notebooks I bought myself while a student. I eventually passed it by my PhD advisor, he made some edits and suggested some changes, helped rewrite the introduction and conclusions, checked the math, and we eventually published a paper together. It was a very satisfying way to bring my side-project to a close. The material never even appeared in my dissertation (as I didn't do a so-called stapler thesis).