Don't do it. Even if you could get the 1071 firmware running in the 1061 it would not give you the same precision as the 1071 can.
First off, the zener diodes that provide the reference voltages are selected to lower standards. Second, there are other components on the analogue board, not just resistors but op-amps, which have considerably lower specs than the ones fitted in the 1071. All you will get is an inaccurate 7.5 digit meter.
As to your firmware problem. I have a 1061A which is a parts mule for my 1071. Which EPROM in the image below is the one that you are calling "the third socket"? (Click for full resolution)

I can remove the stickers from the EPROMS on my parts mule and see what type all of them are. Maybe it won't be useful as my 1061A has date codes from 1984, 85 and 86 on the digital board's ICs.
Sadly this 1061A does not have the GPIB option fitted either, so although I can dump the EPROMS and send you a copy, the dumps might not be any use as from what I understand there is a dependency between the digital board's firmware and the existence of a GPIB interface card.