Hi,
I'm not even entirely sure what the question was and I'm not sure I understand what your ( @paul cotter ) accelerometer is supposed to be doing.
What I'm surprised about is that @alancalverd hasn't mentioned something that he almost always does mention:
To the best of my knowledge, when an aeroplane (and presumably also a spacecraft) is just sat on the runway (or launchpad), without any engines running, prior to any launch / take-off.... the accelerometer in the cabin reads 1g and most certainly NOT 0 g.
It won't say 0 g unless you have chosen to calibrate the accelerometer to display 0 while you were on the runway (or launchpad), which would normally be a completely inappropriate thing to do. In an aeroplane the accelerometer is important for knowing how much stress you can put parts of the aeroplane under in your aerobatics. So it's important to know, for example, that the wings are already being pulled downward and away from their attachment to the fuselage with a force corresponding to 1g even while the aeroplane is just sat on the runway.
I don't know many more details... ask @alancalverd - the aeroplane enthusiast. For example, that 1g that is displayed in the cabin while on the runway is that described as an upward acceleration or a downward accelertaion? I'll assume it's like a conventional accelerometer as described here: https://en.wikipedia.org/wiki/Accelerometer so that this will be displayed as an upward accelertion.
Anyway, this is important because I am not clear about what it is that you ( @paul cotter ) think your accelerometer will be displaying.
....Firing up the engine will produce an acceleration of 9.81m/s and an accelerometer in the craft will register 1g....
If the engines can only produce 1g and the spacecraft was starting on the surface of the planet, then the accelerometer in the cabin doesn't actually move or change one little bit. If I've understood the situation correctly, then it would still read 1g exactly as it did before the engines fired up. The only difference is that the engines are now fighting against gravity rather than the ground that the spacecraft was resting on. The accelerometer in the cabin just doesn't care who or what was countering the gravity and indeed anything left lose in the cabin, like a pen the pilot takes out of his pocket and releases in the air, still behaves the same as it did before the engines fired up, the pen experiences a downward acceleration of just 1g.
Wikipedia defines the term as follows:
...Proper acceleration is the acceleration (the rate of change of velocity) of the object relative to an observer who is in free fall...
So, just from that definition, if the only acceleration a body was experiencing was due to the gravitational field in the region, then that body has precisely 0 proper acceleration. So, well... I'm not sure how to phrase this exactly.... It is apparent why gravity doesn't count as something causing proper acceleration: The 0 point for proper acceleration is deliberately chosen to be precisely that situation where the body experiences what will appear under Newtonian mechanics and Newtonian gravity as only being due to gravity.
There is some confusion about "proper acceleration" but that only exists when and if you interpret it as being an acceleration relative to an inertial frame. If you're using Newtonian mechanics, just don't do this. The definition of proper acceleration stipulates and therefore demands that acceleration was measured relative to an observer in free fall - and as we all know in Newtonian mechanics, an observer in free-fall on or near planet earth is NOT something we would usually consider to have an inertial rest frame. Under Newtonian mechanics and Newtonian gravity we would usually consider an observer in free-fall to be accelerated and therfore to have an accelerated rest frame. To be clear, under Newtonian mechanics (including Newtonian gravity) you simply have NO RIGHT or ability to talk about proper acceleration as if it is an acceleration realtive to an inertial frame (unless you're in the extremely privileged position where the local gravitational field was 0 and an object in free-fall doesn't really "fall" or get accelerated anywhere).
However, you can legitimately talk about proper acceleration as an acceleration relative to an inertial frame when you use GR. This is because GR defines inertial frames slightly differently to Newton's approach.
Newtons inertial frames:
Under Newtonian mechanics, an inertial frame is one where you expect Newtons laws to hold including the law that states an object will move in a straight line unless acted on by a force.
GR inertial frames:
In GR, an inertial frame is defined as a frame that is in free-fall. You throw away some of the things that Newton may have expected. For example, you don't expect or require objects to move in straight lines in an inertial frame when no force acts on them. (As it happens, the main rule governing the motion of such objects is that they will move along geodesic paths, these geodesics will not be straight lines unless the local spacetime curvature is 0).
(https://media1.tenor.com/m/CEWRdC8p87IAAAAd/freefall.gif)
[Image from Tenor.com Seems ok to share when I'm not making a profit. ]
I'm hopefull that this may put some things in some order and perspective. The Wikipedia article about "accelerometers" is, in my opinion, a bit sloppy because it uses the phrase "inertial frame" where only Newtonian mechanics has been used in the rest of the section / paragraph and the reader would reasonably assume that Newton's notion of inertial frame was being used. This is simply misleading, you must use the GR definition of inertial frame if you wish to consider an accelerometer as showing an acceleration relative to an inertial frame. If you wish to use only Newtonian stuff, then a proper acceleration is simply one relative to an observer in free-fall (so that a 0 proper acceleration is almost always some non-zero acceleration in a Newtonian inertial reference frame).
Best Wishes.