I have just looked at the antenna and it is made in 4 parts; the whip,
the chrome spring that the whip gets screwed into, a metal tube [with a
plastic sleeve covering it. could this be the loading coil?] and lastly,
the mag base. A small quantity of RG 58 is soldered to the underside of
the "loading coil", making a very compact antenna.

If I loosen the set screw in the base and pull out the whip, it is 28"
long; I'd like to cut it so that it will receive/resonate center of the
151-174 MHz band. where I do most of my summer monitoring, using my old
[but still working] BC 210.

How long should I cut the whip to accomplish this? Or is the whip too
short already!?

Short out the coil and use length(cm)=7300 / MHz as the total length,
including the bridge over the coil.

You could alternatively use the coil to correct the excessive high.angle
radiation of a 5/8-wave antenna. Use total length(cm)=1860 / MHz including
the length of the turns of the coil. If in practice you find you have less
gain than a simple 1/4-wave then go back to plan #1 above.

Note that these lengths are correct, but reality has a habit of varying, so
cut too long and clip the length for best VSWR.