Why do my LED lights require a different voltage when they are hardwired?
Coin cell batteries have a high internal series resistance. If you measure the battery voltage when the LEDs are on, then you'll see that the voltage is no longer 6V.
Direct from energizer:

For example, the starting IR of a 2032 battery is near 10 ohms

IR = internal resistance
https://data.energizer.com/pdfs/lithiumcoin_appman.pdf

As for your safety question:

Is there a safety concern with the LEDs when hardwired using a DC and step down converter?

That depends on what you are using for your offline power supply.  If it provides some sort of isolation, then it's most likely safe.
There is no inherent reason you can not do this.  It is in fact how almost any LED lighting system works (that plugs into the mains supply).