There are some steep hills, and at one point you have go on a little bridge over a ford (NT 6075 7795) but it is cyclable so no worries there.
The rest is on metalled roads apart from "Becky's Strip' the bit the you get onto after Whitekirk (NT 5892 8194). It's basically a path over a field. There was a plan to make it into a proper bike path a couple of years back but it was shot down by some locals and horse riders who thought it would spoil the ambience or something. Anyway, I think it was surfaced with some acceptable material, but not having actually ridden it I'm not sure about the suitability for your bikes.
Have a look at and post your query on the Dunbar Cycling Group's facebook page, others there will have better knowledge than I.
https://www.facebook.com/DunbarCycling