In regards to Cawley, the source he cites is the Chronicle of Roger of Hoveden, volume 1, p. 59, which is here.
It's in latin and my understanding of latin isn't that great, but it seems clear that Roger of Hoveden, is using Simeon of Durham as his source, and he is only saying that Gospatric is the son of Maldred who is the son of Crinan (Cospatricus filius Maldredi filii Crinani, and Maldredo filio Crinani Tenuit). Like Simeon of Durham there is no indication that Crinan is the same as Crinan, father of Duncan, and more importantly states nothing about Maldred' mother.
Cawley's statement that Bethoc of Scotland is Maldred's mother isn't supported by any citations at all.
I'll have a look at the merges.