You would need to find an EA53 tube & spring to use that probe, some 410B probes only fit one type of tube too.
The EA53 has a 6.3V heater, 2-01C has a 5V heater, the 410C has no adjustment for this.
If the 410C was mine I would look for the correct probe and save that NOS probe for a 410B, both types of VTVM in good working order are quite pricy these days.
Did you compare component date codes with the A3 board & the rest of the 410C, to determine if it is original, or a repair replacement due to failed neon photo chopper assembly?
It isn't clear from the two manuals in the Keysight (Agilent) pdf, at which serial number the change was made, note the A3 tube heater was fed from the +6V dc supply.
I do have pictures of one with the optical chopper board, that otherwise looks the same as yours, but the serial number plate had been removed, quite strange as it looked in almost new condition.
David