I dislike staining/dyeing wood too (even more after this experience). If I knew that sapele with an oil finish would have make the recipient happy, it wouldn't have ended up costing as much as I first thought. I've never used it, so I didn't have any samples to show her.
Barb, yes, that's the exact shellac I had hoped to use. The reason I didn't is this section of the Bulls Eye Shellac data sheet.
I bet it would have worked just fine, but I didn't want to risk being wrong.
Splint, I didn't expect any problems using TransTint in the water-based poly because the TransTint directions indicate it should work fine. TransTint's customer support didn't say anything different either. But it just seems to me that it's incompatible with this particular poly. (Or I just had too much dye.) If I had some GF gloss/clear poly on hand, I'd try it to see if the flatteners were the culprit.
I have mixed feelings about the finish I achieved. There is some splotching, and the finish looks great in some light and sort of dull and washed out in others. But our friend loves how it looks, and that's the important thing.