I hear you and understand your concern. As Peter Drucker states, the first duty of managers is "to strive for the best possible economic results from the resources currently employed or available." Good managers like you work hard at effectively monitoring and controlling costs. Where you may have missed the mark is your summation of the upside. The benefits of improving culture and, consequently, employee engagement is much more than making people happy.
Creating a culture conducive to people being passionate about and committed to their work and their organization provides improved performance in several areas (financial, safety, absenteeism, shrinkage, retention, and customer satisfaction). Said differently, engaged employees are: more productive, generate more revenue, have fewer injury claims, fewer lost workdays, and fewer injury costs, don't steal, make fewer errors, stay at your organization longer, and produce more loyal customers. Hopefully, you currently measure each of these areas as they relate to your organization. If not, here's a helpful tool for helping you begin measuring and quantifying the impact of underperformance in these areas. (The article relates to the safety industry but the algorithms are suitable for in any industry).
If your concern is genuinely related to costs, looking at the numbers is a great way to determine if improving culture is a worthy investment. If, however, your concern is about investing in your people, know that research has shown that especially in unprecedented times, companies perform better and are shielded from the results of high recessions when they invest in their employees and their skills and weather the storm with them. The law of reciprocity is well-proven. When you do the right thing and take care of others, more often than not, they will take care of you.