LDAP authentication can mean different things. With RStudio Server, your Linux system is connected to LDAP, probably via sssd, and takes care of creating a Kerberos ticket during login. Since the R process executes with your user rights, it can use the Kerberos ticket for connecting with the database.
RStudio Connect connects directly to the LDAP server to authenticate users. These users (normally) don't exist on the underlying Linux system. And the R process supporting your Shiny app is run using the user rights of some technical user.
As for solutions, you have to first answer one question: When two different users access the app, should the data returned from the database be the same or should it be different, since these two users have different access rights on the DB? In other words, does the data access need to use the user's credentials or can it use the credential for some technical account?
Some relevant links: