Let them be emotional.
You do risk losing your client because there are many in this game that give away stuff all the time but there are a ton of downstream threats caused by that that negatively impact both the client and the ones giving free stuff away. That said, if they find your work valuable, then likely it's just a tantrum to try to get their way.
There are other clients out there. Why degrade your profit margin and take on other risks for a client when you can replace them with another and maintain a healthier profit margin.
EDIT: There is a presumption in projects that change will happen and that it will impact cost, price, and time. So you can approach this with an 'assumptive close,' that you assume the client already knows there will be a cost impact and, therefore, you are merely telling them how much. The "no" is implicit.
Validate the efficacy of the change, describe the cost and schedule impacts, state your company's position and then simply let them throw their tantrums. You never have to curtly say no.