How can I prevent my multi-meter fuse from blowing with a brief 120VAC surge?
I think you're barking up the wrong tree.
You are attempting to view a momentary surge with a common DVM.  They're not made for that.
You would be better off installing an ammeter shunt, and then looking for millivolt rises across the shunt.  Further, you should use some sort of data recorder that will sample at a very high rate, so you can observe the transient.
Most likely this was the GFCI device's internal electronics initially charging their capacitors. It could also be some part of the startup testing that some GFCIs do.  For instance some are able to detect neutral-ground faults on power-up; they can only do that by attempting to flow some current on neutral.  Whatever it is has to fit inside a recep form-factor.