I remember as a teenager in the machine shop there was a Delapina Hone (American possibly?) with expanding mandrel(s) for lapping small bores. Used (by me) for sizing small ends on car con-rods after fitting new bushes (Bronze, made and fitted by me). I learned a lot working there. My father made his lap (I still use it) from a piece of broom handle (hard-wood) with a groove, springs, and 1/4" square stone slip fitted. His standard bore was 1". I remember talking to him about whether it would taper the bore, and as he was only taking the finest polish lap he explained "possibly, but not measurably, by him". It was a prototype, as he planned a mild steel version with setting screws to set the stone parallel, and for progressing the cut. But that did not become necessary.
Before I inherited that lap, I used a piece of wood dowel with layers of paper taped-on, using grinding paste on the top layer of paper, or with the top layer being 1000 grade wet and dry paper (oiled). With the paper very tight, you can increase the cut by adding a single layer of paper (+0.002" radius?) if not adequate pressure at the first attempt. It takes a lot of care and fiddle to get the cylinder fitted onto the lap, but remember, you should only be taking the peaks off the bore machining, not actually removing metal in any measurable amount. The finest grinding paste I have used (on brass cylinders) is toothpaste. ALWAYS thoroughly clean all bores after lapping. (And after they have written long texts like this one!). I think the paper lap may have "belled" the ends of the bores slightly, but I couldn't measure anything. The piston just didn't feel so tight a sliding fit at the ends... But I had the same effect when using a bought sprung lap. As a teenager, on car bores, with a tolerance of 0.001" max total variance, care was always taken to prevent any taper or belling of the ends. (DTI measured 0.0001" intervals. most bores were within 0.0004~5" variance measured in about 10 places anyway. Compared to typically 0.004" wear (or more!) on bores before we did the re-bore, it was good enough).
I am sure Brian can make a good lap anyway.
K2