A couple thoughts for you...

You describe the tubers as "small congested clumps of (pot-grown) tubers," which makes me think that it might be difficult for even an expert to successfully divide them. Many growers would just keep the pot tuber planted intact in a potting mix, take cuttings until enough plants are created, then plant out the clump as-is.

If you really want to divide it and are thrown off by the roots, it is a safe bet to divide directly down the middle of last year's stem, being sure to avoid slicing the necks or bodies of the tubers, and 'wiggling' the two halves apart. You might lose a tuber or two, but there should be eyes on both halves if you do it this way.

I don't personally know any growers that use straight peat moss for prestarting tubers. I would be concerned that it might hold too much moisture next to the tuber and invite rot. Some combination of peat, vermiculite, perlite and/or 'sharp' builders sand provides better drainage to the peat. (Growers in the UK especially feel it is important that sand be in the mix, although I have never used it and don't quite understand what benefit it has to the tubers or cuttings. I have asked a couple UK master growers, and they respond 'drainage' and that it always works for them)

Hope this helps... Good luck, and let us know how it works out for you!

Cheers,

CiCi

I've not had any experience with taking cuttings, so I'll go for planting the whole pot tuber/s intact. Last year I planted some Bishop of Llandaff in open ground. By the end of the season when I dug them up, the clumps of tubers resembled an Octopus in shape. All stuck out sideways, and not like the vertical bunches one sees in packets for sale.