I think in 1990 there was an SEF/trigger group grip that locked out the FA position until a button was depressed. Required 2 hands. RTG had these for sale at one time sometime back and I believe they were referred to a police grip/trigger group on that site.
In answer to
Is there a certain name you have for the mp5's that have the button you need to push before you can move the selector to full auto?
HK Germany marketed the MP5Z series for the West German police forces with a selector button on the right hand side that you had to push in to move the left hand side selector lever from E to F.
The Z stood for "Zusatzsicherung" or "special safety (device)".
I have seen a brown colored manual for this in German dated August 1986 with illustrations for the standard fixed and sliding stock versions (MP5A2Z and MP5A3Z), the K versions with and without diopter sights (MP5KZ and MP5KA1Z) and also the suppressed stocked version (MP5SD2Z). All are shown with scope mount blocks.
[On MP5s more generally, German versions will have HK and the Ulm proof house stamped on the top of the receiver but Turkish (MKE) and Pakistani (POF) will not. I have seen POF versions marked e.g. "MP5-A3" which I have never seen on a German model (which would simply be marked "MP5" for the standard A2 or A3 versions).
The outer hand grip is the same SEF type as the standard MP5 or K but the metal inner box which holds the internals has a special slot to accommodate the Z safety lever so you cannot just drop the Z lever into a standard SEF housing/grip and expect it to work as intended. (An ordinary selector will function fine in a Z grip).
HK then changed to a lockout safety selector lever that needed a special tool/key to move from standard operation to locking out full auto and back. That selector lever was designed to be dropped straight into any SEF grip.
These systems were perhaps a solution to an unnecessary problem since IMHO sound training ought to answer this type of issue. Purely political points like "not full auto" are answered by the FBI grip. (My 2c anyway)