"Peter R." ...
The vacuum gauge is showing 4.5 inches
or so, which is steadily in the green.

OK, many A&Ps tweak the vac regulator toward the higher end of the range to
get a few more years out of a lazy gyro. Yours is running at the lower end.
Another bit of info you should have at hand (search in your logs) is when
the filter was last changed and the age (hours) of your Vac Pump. This will
save your mechanic the time to look it up.
Then you or your mech should get a good look behind the panel to make sure
the vac line for the AI is not pinched or kinked (sometimes happens at
install or after other work in done). This would make the AI get only a
fraction of the airflow needed.
Finally, keep a really close eye to see if the AI ever "hangs up" (even for
a second) once it comes alive after engine start. A "slow to awake" AI is a
annoying, any AI that "stops or stutters" once taxi or flight begings is
broken and downright dangerous.