@john_mallord,
I was going to suggest placing the resistor on the PSU board to bridge a gap cut out of the top trace. The doubled up grounding trace means you'd also have to create a gap directly underneath in the bottom trace before drilling a couple of 0.6mm holes for a 1/4W 1 to 10 k wire ended resistor, ideally stood off from the board by 3 to 4 mm with ceramic beads.
In view of this extra complication (and the more permanent modification of the PSU board), a better, reversible option would be to place the resistor in series with that black earthing wire. That way, if felt necessary, you can at least undo the modification without leaving any obvious signs of it ever being attempted (unsolder the wire from the PE tag to insert the resistor into circuit).
As to the question of powering it from a battery, a good starting point would be Bad_driver's earlier contribution to this very topic thread right here:
https://www.eevblog.com/forum/testgear/re-feelelec-new-arrival-fy-6900-signal-generator/msg3836159/#msg3836159 Since you only need a single 5v 2A max supply, an alternative option could be to use one of the later versions of battery banks now available which can act as a mini 5 volt UPS allowing you to wire it up between the existing 5v psu and the mainboard via a separate 2 pole change-over switch to save discharging the battery when not being used.
This would allow you to disconnect its output and allow the existing PSU to power it directly as before whilst still providing charging current to the power bank. However in this case, I think you may need to upgrade the PSU to a 3 amp rated one to avoid overloading the original in a worst case scenario of low battery state charging demand (about an amp) on top of a worst case demand from the main board (possibly a maximum close to the 2A limit).
An alternative to this arrangement would be to use a separate usb socket for charging the power bank and use a single pole change-over switch to select between battery or mains power operation, allowing you to retain the original psu board.
Since the psu board is a class II double insulated unit, no safety earth connection is required. For our purpose we really only need access to a PE tag to connect a relatively low resistance "Static Drain" resistor to attenuate the half live mains voltage down to a vanishingly small value (0.25vac or less with the suggested resistor values). Even a drain resistor value as low as 1K will attenuate the unwanted noise and random DC offsets, polluting the mains earth wiring, by some 60dB or so.
'Scopes, otoh, tend to use class I psus which demand an actual protective earth connection and it's generally considered not a good idea to disconnect this protective earth in order to eliminate any such ground loop issues, even when this is only a "temporary measure".