I can give you the OFFICIAL answer and then my ED answer.
The OFFICIAL answer is:
The Exercise Trail is, as you say still off limits to all traffic.
The Bus road that runs parallel to the ET (West Wilderness Road) is off limits to runners.
The main entrance to the Fort will not admit you if you are not a guest there, not entering to visit a specific family name at a specific family campsite/cabin, or if you don't have a revenue generating reservation (HDDR, archery, Segway tour, etc.) within an hour or two of arrival.
The unofficial ED answer I would offer is:
The Fort internal bus lines between the HDDR-end (Settlement) and check-in end (Outpost) start to run at 6am (infrequently in the first hour or so). Those lines include a BUS from Settlement (near HDDR) to Wilderness Lodge (for folks looking for table service breakfasts, for example) and then return back to the Fort and run an internal route.
At WL the bus stops have this sign and you will want a bus in slot #1 that is running to FW (buses in this slot serve MK and FW in both directions). Although at an early hour like that it's not headed to MK or back from there yet.
But if you are at the bus stop at 6am, a bus from FW will pull in soon and then leave out to return to FW (with you on it). That means starting soon after 6am but it should get you there.
I have a favorite route I walk/run at the Fort that is a 5k/3.1mile route which you could do twice (maybe in opposite directions?) to get your mileage in. It follows the two big main roads (Fort Wilderness Trail and Big Pine Road, adds in the main road through the cabin loops (Peacock Pass), and then throws in the two camping loops 1500 and 2000 get get the 5k distance right. Or leave off the loops 2000 and 1500, just follow the main roads, and it will be 2.9-3.0 miles.
View attachment 914352
Spanish Moss Lane is loop 2000, btw. The bus from WL will come into the Settlement bus depot which is at the red dot at the top near the HDDR symbol.
The nice thing about this route is you can run on the main roads or the close-by sidewalks if there is vehicular traffic. Finish back where you started and take a bus from the Settlement bus depot back to WL and cool off.
The sun rises to the east (right side) of these maps along Big Pine Trail and I am always struck by the silhouette of the trees here as I'm out early in the morning.
That's my suggestion. The Fort is a great place to put in some mileage but access before 6am would be difficult.
Bama Ed
PS - you could try to talk your way in the front gate at 5am but that would be a little hit-or-miss IMO.
PPS - also at the early 5am hour, the West Wilderness Road between WL and FW won't have much/any traffic anyway. And you could always take an internal Fort bus back down to WL from the Settlement at the end of your run after 6am.