Yes, the flight of steps up from the canal to the climbing centre is far from bike friendly, despite having a (steep) wooden "bike gutter" alongside it. I certainly wouldn't even contemplate trying to manhandle a cargo bike up it.
Coming from Edinburgh, I think you should be able to continue past the steps, under the road bridge, for about 1km or so until you reach Wilkie's Basin (GR NT119710), from where there is a track that runs alongside the field to the north of the canal that takes you to the nether regions of the climbing centre car park. I'm not sure of the actual state of that track, or how many gates you might encounter requiring you to dismount to open them. I've only eyeballed it from the car park end. It looks OK-ish on Google Maps satellite view - I wouldn't hesitate to give it a go on my MTB - but your cargo bike might be more of a handful if there are any significant ruts/potholes?
The other way would be to come off the canal in Ratho village, head north past the church and then take the track signposted "Ratho Paths Network Wilkie's Basin via Platt Hill". I'm led to believe that this track is in reasonable nick now (it certainly wasn't when I was taken along it in a big 4x4 for a sneak preview of the climbing centre before it opened, but that was a good many years ago now) and again there may be one or two gates to negotiate. This track is marked as a route to the climbing centre on Google Maps.