Any suggestions on the best way to get from Edinburgh leaving September 4 arriving Canterbury on September 5?
By train, one change in London required.
You can easily do that on the train in a day. There is a train every 30 minutes which you might pick up for £38 if you book well ahead, more typically £50 to £60. It takes about 4 1/2 hours to London Kings Cross.
Two companies compete on the route-Lumo and LNER.
Walk literally across the road to St Pancras Station and get a High Speed train which takes 1 hour to Canterbury, £14 if you book well ahead on off peak trains (off peak ends at 1537 on that route and restarts at 1900). So leave Edinburgh around 0900 and you'll be in plenty of time to beat the peak hour trains out of London.