I get what you mean. Depending on delivery, those phrases can convey irritation. But if you make sure to sound ultra sincere and sympathetic (conveying 'warmth'), it will come across as if you're really serious about your work and care about the other person. In this sense, you could definitely use 「安心してください」、「（それに関しては）安心して欲しいです」.
「ご安心頂ければと存じます」and「ご安心頂けましたら幸いです」are super super polite. These do not convey irritation (unless you are, in fact, irritated, and the other party can tell this by the tone of your voice). The latter of these two is 'softer' in a sense and could therefore be better if you use keigo.
「ご安心ください」is also a good choice. This is strictly speaking not as polite as the above, as you are "requesting" that they feel relieved, instead of saying something along the lines of "It would make me very happy if you were to feel relieved".
「ご安心を」is great if you don't have to be ultra polite, especially if you can say it with a deep, calm, resolute, professional voice. If you're not confident in your delivery though, using the super polite ones with a sincere tone should always work to some extent, and in that sense I would recommend those, instead.