Your medical if performed by an AME will show two dates one for expiry of the class 2 and the other the expiry of the LAPL but it is the licence you have to go with them that is important. So if flying on a LAPL medical you will require a LAPL licence although a class 2 medical will also do. For a PPL you will need a class 1 or 2 medical.
Not sure why you say that getting a medical for LAPL and Class 2 at the same time will save money? Only an AME can issue a LAPL and a class 2. Your GP if you have no health problems can issue a LAPL.
If you plan on upgrading to a PPL later the CAA will take more money in fees! So factor that in to your costings.