I can tell you, as it applies to military bases, "other lawful purposes" is specifically defined by the Commanding Officer of that base and will be written in an instruction. Some bases allow hunting. Some bases have gun ranges on them open to military members. Some bases have family housing units on the base.
The Commanding Officer is in charge of his/her base and they set the policies, for the most part, about who gets on that base and what they can or cannot bring with them. Most of the time, when you get to the gates of a military you will see a sign that says something similar to "all firearms must be declared prior to entry". That is when, if you did not know the policies for that base, you would declare the firearm to the gate guard and, hopefully, they would know what would be required for entry or not.
Unless I possessed some sort of written authorization to bring a firearm onto/into a specific Federal facility for a specific purpose, I would not carry past an 18 USC 930 sign on a state issued permit.
And, here's a general rule of thumb anyway, that I just thought about. How can a state AUTHORIZE anybody to do anything on Federal government property anyway? If the Federal government bans something at the Federal level - a state cannot "unban" that item. For instance, Florida could not "unban" Cuban cigars and make them legal to possess in Florida. So I don't see how a state could overrule the Federal government by giving you a state issued permit.