If I only have room for a short dipole, say 40 feet, and use a tuner such
as the SGC or Icom AH-4, what is the penalty in DB for the lower bands
(160, 75, 60, 40) compared to a half wave on each band?

Some quasi-wild guesses for a 40 foot dipole relative to full size at
the same height, probably best case.

160m: -16dB
75m: -8dB
60m: -3dB
40m: -1.5dB

These numbers don't mean much. I used EZNEC to look at the losses in
Q=100 loading coils with enough reactance to bring thing to resonance
on each band. No attempt to match to 50 ohms was made, but I made the
Q kind of low to reflect other tuner losses. Still, these are almost
certainly optimistic, especially on 160.

Dan