Good morning Greg
Impressed by your measurements of the peg. You have identified slight wear (a bit over 5 thou on diameter) and there will be corresponding slight but acceptable wear on the cam. Just as Bishop expected and for which they provided compensating adjustment. You will rotate the peg which will have long-term benefit but will not, I think , do much for 5"+ of free play. You say the shaft looks fine and diameter measurements would confirm this.
You are now intending to remove the steering box to 'rebuild' it. The implication is that you are of the opinion that the cause of the free play is in the steering box rather than in the adjustment of it. But where is the evidence? No opinion is worth holding unless it is supported by evidence. The only evidence I am aware of is your confirmation of no axial movement of the column (hope you had the wheels off the floor). This suggests there is nothing much wrong with the cam/column assembly, the upper and lower bearings or the shimming. Now this is not decisive evidence but it is negative rather than positive and supporting your opinion. And what would a rebuild involve? A new cam and new bearings? Hardly 'off the shelf' stuff. I think Jonas may have researched this, or have knowledge, and his advice would be helpful. I have in mind a lot of time, a lot of dollars and replacement parts of questionable quality. Sounds to me like a high cost to benefit ratio. But it is your car, your time and your cash..........
And what about adjustment? Seems to have gone out of the window. With respect Greg, I think you are going the wrong way at the fork in the road. If I were you I would refit the rocker shaft in the straight ahead position, crank and peg at 90 degrees to the axis of the cam. No need to replace the drop arm at this stage. Fit the Bishop cover plate and adjust the box, wheels on the floor. Somebody banged on about this at great length above. Only if careful adjustment failed to eliminate the free play would I think 'rebuild'. If you take the other road at the fork - rebuild, fit the Bishop cover, adjust and eliminate free play you will never know whether all those dollars were wasted and you will have concerns, perhaps, about the quality of the replacement parts. If you doubt the wisdom of my advice, ask your wife for her opinion as she is clearly a member of 'Team 59 Aceca'.
Now, to cut to the chase (I do go on a bit). You asked for advice about removing the steering box. Unless you have evidence (sufficient, relevant and reliable) that it is broke don't fix it. My advice is to leave the steering box where it is and adjust it.
By the way, I take issue with your description of the Burman steering box as 'antiquated' (OED = old fashioned, out of date). It was good British engineering in the 1950's and still is in 2023. (Pause for 'Rule Britannia'!)
Regards and a Happy New Year to you. May 2023 be the year you get your Aceca steering safely, indeed as it did ex-Thames Ditton in 1959.
Michael