I've always been concerned that Amsteel or equivalent will not stretch and thus provides no shock absorption. I've always used 1/4" (5mm) to 5/16" (6mm) polyester, depending on the tramp size, and have been pleased. Normally I replace the lacing about 70% through the tramp life. I've never had any trouble with the polyester losing tension.
I tension with a pair of vise grips (to hold what I've got and pause) and an acender (similar to those used to climb a halyard
, to pull the line). Very easy, nice handle, and you should have one.