Possibly! On my Vienna's the barrels do not have a second hole, so the only way to record them is to disassemble the movement, disassemble the great wheel from the barrel and then poke the new cord through the small hole, tie a big knot and pull tight then put it all back together again.

If you can get at the inside of the barrel through the 3mm hole, then you may be able to tie the knot and pull it inside. A "leader" of soft thin wire which is easier to manoeuvre and bend to capture through the 3mm hole may help. Measure the length of the original cord and tie a loop at the other end to hang on the hook on the baseplate.
Once reinstalled in the case wind about half the cord onto the barrel while keeping some light tension on it, thread on the pulley, hook the other end in place and add the weight. Then wind up the rest.
If you fully wind the cord onto the barrel before hooking the other end up, it all gets difficult under and behind the dial. Half way gives you room to work.
I would change both (and if you can strip down and service the clock) while you are at it – once one has gone, you will be doing it again soon for the other.
Removing the dial and hands probably won't help unless the front plate has holes in them through which you can reach the barrels. Most Vienna's don't. Just be careful how you lay the movement down to work on it. I use a plastic bowl about the same size as the dial to make sure that it's not resting on the hands.
Typo
Edited By Peter Cook 6 on 06/08/2022 18:44:30