If you knew Barry, you would not think that

If Barry said it would, and you know Barry, I don't think I would bother asking
anyone else.

The question still stands - would adding a 4:1 balun at
the transition between coax and ladder line improve the
system or is a coax choke good enough?

I assume you are to use this antenna on all bands. The reason for adding a
choke balun is that you are going from an unbalanced coax to a more or less
balanced antenna system. The choke attempts to maintain that balance by
presenting a high impedance path to any current that attempts to flow on the
outside of the coax shield.

Adding a 4:1 balun might help the tuner see a more easily tuned (less tuner
loss) impedance on some frequencies. However, this is an all band antenna, so
the Z is all over the place. A 4:1 impedance transformer most of the time
won't help, and can even make things worse. I wouldn't bother.
If the tuner balks on a certain frequency, change the length of 450 ohm line
a little till she will tune. About that 25' of RG-58 between the tuner and the
450 ohm line. (8' to the window+17' in the choke) That will transform the Z at
the 450 ohm termination to something completely different at the tuner. The
exception is that the Z at the 450 ohm line is 50 ohms resistive, then it is a
true 1:1.
You can see all this stuff in action if you model your antenna and
transmission line on Eznec, or Reg Edwards Dipole1.


73 Gary N4AST