Sir Michael Parkman . . . . .
With your mention of somewhat of an extended use time.
With the unit unplugged, check my two wires at the pointers that go to a thermal temp switch mounted against the power transformer core.
There is ALSO the thermomechanical switch which makes a noise . . .not of our current concern.
In your case I think that you are dead in the water before even reaching the transformer secondary, because if the item that I mentioned having opened up, therefore . . . .no power to the transformer primary.
If your transformer doesn't smell / stink like an overheated old inductive fluorescent lamp ballast . . . .and I don't think that it will . . . you will still be in business, with the original power transformer.
And I think that my dealings with 5 Schumacher units amongst our families, within long time periods, I do find them to be quite hardy..
If confirming that fault, initially use 2 clip ended test leads to get your DVM placed in 10A AC amps configuration to see just how much AC line power amperage is needed at a hefty 6 amp load on your secondary battery side.
Run the unit awhile to confirm no odor emanating forth from the power transformer . . .which I don't expect.
Then use the shown fuse clip/holder and a standard fast blow fuse rated at about 1 1/2 the measured primary current pull.
Then give that unit a shake down on the safety of a concrete floor.
If successful and getting long runtime, but possibly wanting to replace the thermal fuse, contact me for more specific how-to details .
Or come back if any more ?'s.
Your Schumacher . . . .
73's de Edd.