There are always risks. The real question is which approach gives risks that are more acceptable to your situation.
The use of LDAP for this is a tried and trusted solution that is well used and well supported. So that is a definite positive. Your bespoke auth is potentially poorly tested and, well, bespoke! How do you prove the risks associated with it? That might be unknowable without spending money on getting it professionally tested.
I would say though that there is LDAP and then there is LDAP! If they really mean that you would be using Active Directory along with the Microsoft standard tools for authentication and authorisation, that makes a whole load more sense than if they are simply saying "let's just use LDAPS".
Vanilla LDAPS may be no better than your current solution. But it might still have advantages if the current solution has minimal support and maintenance.
Bottom line is that off-the-shelf solutions, especially when using well tried industry standards are almost always better than bespoke ones for standard tasks. And are likely to be better supported and tested. But adding a specialist toolset is likely to be even better. Use open standards wherever possible as it makes integration and maintenance a lot easier. So using OAuth for example.
Incidentally, LDAPs isn't difficult to set up. Active Directory can be difficult if you don't already have a Windows infrastructure. If you are already using AD in the same environment as your application and users, it is a bit of a no-brainer.
Assess the risks, impacts and benefits of each without the emotion of protected territories. This should be a documented risk assessment, probably with a benefits matrix.