Firstly, 'Qingcheng Mountain / Mount Qingcheng' is partly a transliteration (I believe the full translation is 'Six Senses Mountain'. While perhaps evocative, this is nowhere near as mellifluous as 'Qingcheng Mountain', which is the better choice.)
Secondly, neither of your options sounds too attractive (this is not saying they're 'incorrect', but a reasonable paraphrase might well sound better).
Answering your actual question, neither order would be unacceptable, though it's unusual to place an adjective (outer) after an attributive (or grading to compound) noun (Qingcheng). Lower Yellowstone Falls follows the usual pattern, but Newton Lower Falls doesn't. It probably depends on the relative cohesivenesses perceived between say
Lower..Yellowstone,
Yellowstone .. Falls, or
Lower .. Falls; and the complete expression
Lower .. Yellowstone .. Falls.
There is evidence that this is so in the different usages
It is also suggested that there could be an outer London ring road
beyond the M25 [BBC News
2002]: outer / London
('outer' to the existing one) and the more usually found (but note the attributive ... adjective order)
The M25 London outer ring road [An Irish Tail: A Hilarious Tale of an English Couple and Their Unruly Dogs ... Nick
Albert
]: London / outer
Once expressions, particularly proper nouns, become fixed, it's fairly pointless trying to explain whether or not they are as rigorously grammatical as some would like them to be.
I don't really feel competent to judge which (if either) is better in your case. But I would say that translations of Chinese place-names often sound a little quaint to Western ears – but that just adds to the charm. If it has to be one of the two, I like the sound of 'Qingcheng Outer Mountain' slightly more than that of the alternative, but that's just my opinion. If the heights are right, 'Little Qingcheng' follows a fairly common pattern in the naming of subsidiary peaks and is quite mellifluous.