NavList:

BillB wrote: Not going to happen. Center of a sun holding at 0 degrees declination
all day viewed in a vacuum adjusted for variation--maybe.

I think the key words here are ‘compass’ and ‘check’.  We’re not talking about an astronomer in an observatory trying to detect a change in earth rotation, or a weapons officer trying to set up an inertial platform to 1/10^th degree or less.  We’re talking about an 18^th, 19^th, or early 20^th century seaman, mid ocean, possibly with an unreliable timepiece, attempting to check that his trusty compass is not taking him miles off course.  If the helmsman could only estimate his course made good over the last hour to about half a compass point, then an accuracy to one or two degrees for a compass check is acceptable.
Most of the sailor’s guides have tips as to where the Sun should be when its true altitude is zero after allowing for refraction, dip etc.  Norie’s Tables say the Sun’s lower limb should be half a diameter above the horizon.  If you prefer to take the bearing when the Sun's centre crosses the horizon, Norie’s give a secondary correction table.  At 47N, at the equinox, this would be 0.7 degrees.  They also explain which way to apply the correction.
It’s worth thinking how to do the compass check.  In extremis you could turn to place the rising/setting Sun on the bow or stern, but this wouldn’t be very accurate, because deviation is a function of the ships heading.  The ship should maintain course and a relative bearing taken on the Sun.  This might be with a pelorus, if fitted, or by a horizontal sextant angle wrt the ships head or stern.  If the compass was the hand-held variety, that could be used directly, as long as you stood close to its steering position.
The difference between the Sun's compass bearing and its true bearing from the book would be the sum of deviation plus variation.  Unless you wished to record each separately for their Lordships of the Admiralty as many early RN explorers did, you need go no further.  You can correct the magnetic course to steer by the helmsman, and have a corrected true course to plot with.  I’m sure Lecky had something to say about this (He did about most things), but I can’t find it right now, and I don’t feel like wading through all 800 pages.
I feel William Bligh might have used this technique during the launch journey. If anyone has access to a copy of Hamilton Moore’s dated around 1780, perhaps they could check for me if they contained Amplitude Tables.  Thanks. DaveP