Most of grammar books describe that due to acts more adjectival. We all know but this question (+1, of course) forces me to dig in deeper. And, I found something useful...
In the book Woe Is I The Grammarphobe’s Guide to Better English in Plain English, the author proposes to substitute "due to" for “caused by” or “resulting from.” She explains that if a sentence begins with “due to,” as in “Due to inclement weather, school was canceled,” the sentence is “probably wrong.” - Grammar Girl
While keeping general rules aside for a special case like this (adjective ahead), I think following this rule does not harm the structure.
Having said that,
Climbing that mountain is difficult because of its height -sounds preferable to me.
Note - COCAE shows results of both the usages (...difficult because of... and ...difficult due to ...) but then the former returns with over hundred results, the latter sticks around a couple of dozens.