OK, forgive me, I'm finding my feet at this depth.
My database prefix is "dbb" not "wp" and the tables I was searching for were on the second page of the tables list, which is why they didn't show before.
the bdd_site table shows only two sites, both their own domains are stipulated and the path both show to the root (this is identical to the setup that was on my old server)
the bdd_blogs is again identical to how it was on my old server.
In the domain column they all have "mysite.com" and in the path column they all have "/mychildsite/" which appears to make sense and be correct to me.
I then looked at my .htaccess file
(reminder: what has been happening is that I can go to my "childsite.com" as it's apparently mapped and shows perfectly well. If then try to go to "childsite/wp-admin/ it tries to redirect to the login page and then eventually to "http://totalsoccerschools.com/cgi-sys/index.html"
I cleared my .htaccess out and put in the min required for sub folders based on this
when I tried this time to go to "childsite.com/wp-admin/" the result was something completely different and unexpected. It redirected me to
This reminded me that I have a redirect plugin for the buddypress homepage, so I could assume that something was associated with the plugins.
I renamed the plugins folder and tried loggining in to the dashboard of my childsite again and I got in.