Greg, I've just been through this with the Mentmore button-filler shown in my avatar. So ...
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What side of the rail bar do I Put the sac? I assume it would be the side with the thin strip which I imagine to bow under pressure from the button.
Not sure what you mean by "thin strip" but you should place the sac by the side that's completely smooth, to avoid cutting into the sac.
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which ever side I put the sac, when I screw the section back in the sac catches and twists with it. This results in a screwed up sac inside.
Two possibilities ...
One, the sac you are using is simply too large; go one or two sizes down. (You could also consider coating the sac with talcum powder or powdered graphite as a dry lubricant.) Turn the section SLOWLY and GENTLY as you screw it all back in.
Two: If your Burnham is anything like my Mentmore or an original Parker Duofold, you should be able to remove the button from the end of the pen. (See my pic for ref.) Just gently pull it out. Then refit the sac/section. Then CAREFULLY slide the rail back in (flat side towards the sac!) through the "button-hole", wait untill it locates against the section, then refit the button. This is the way "Da Book" recommends to do it. I haven't tried it myself, having used the first method. The danger with this method is that you pierce the sac with the end of rail.
Hope this helps!
Richard