Solution 1:

I think you are right that this word was meant to draw attention to itself. It amuses. The entire article is quite tongue-in-cheek — or even sword-in-cheek. Or perhaps even something-else-in-cheek, given how it starts and ends:

Sex and sword swallowing beg some pretty obvious comparisons . . . and exactly how you get something so long and hard down your throat.

Those bookends set a clear tone here. We know what is being suggested.

No dictionary contains all words — nor ever shall nor can. There are many, many different reasons for this. You probably didn’t find flowing-haired in a dictionary either. But because it is created using a productive suffix, everyone knows what it means.

There is no verb involved here, neither to flowing hair not to mustache. The ‑ed suffix can turn both verbs and nouns into adjectives. Think of it as two difference suffixes. This is not a past participle.

Obviously bemustached is an English word, since it was used in English and everyone recognizes what it means. Everyone would similarly know what a bepimpled teen-aged face is, even if you cannot find that word in this or that dictionary.

The construction be- + XXX + ‑ed is a productive way to create new adjectives in English. According to Oxford Dictionaries Online, sense 5 of the be‑ prefix is:

(Forming adjectives ending in ‑ed) having; covered with: bejewelled

So the fellow “has” a mustache, or even is covered with one.

The Dictionary has a lot of these beXXXed words, but not all coinages are befated to be found in a dictionary. Since this construction is productive, anybody can produce new ones of these whenever they please. It might catch the reader’s eye, or the listener’s ear, but this is not a bad thing. Surely this is what the article’s author intended.

Here are some from the OED:

  • beblubbered
  • becalmed
  • beclogged
  • beclouded
  • becrossed
  • becurled
  • bedabbled
  • bedangled
  • bedarkened
  • bedaubed
  • bedazed
  • bedazzled
  • bedecked
  • bedevilled
  • bedimmed
  • bedizened
  • bedogged
  • bedraggled
  • bedressed
  • bedusted
  • bedwarfed
  • befeathered
  • befogged
  • befooled
  • befringed
  • befrogged
  • begilded
  • begirdled
  • begodded
  • begored
  • begrimed
  • begrudged
  • beguarded
  • beguiled
  • behanged
  • bejaded
  • bejewelled
  • belaced
  • belauded
  • beleaguered
  • belighted
  • beliked
  • bemangled
  • bemasked
  • bemazed
  • bemired
  • bemocked
  • bemoistened
  • bemudded
  • bemuddled
  • bemuffled
  • benighted
  • benumbed
  • bepatched
  • bepearled
  • bepitched
  • beplastered
  • beplumed
  • bepowdered
  • bepraised
  • bepuffed
  • bepuzzled
  • besainted
  • bescattered
  • bescrawled
  • bescribbled
  • beshadowed
  • besilvered
  • beslaved
  • beslavered
  • besmeared
  • besmirched
  • besmoked
  • besmottered
  • besmutted
  • besnowed
  • besoiled
  • bespangled
  • bespattered
  • bespeckled
  • bespectacled
  • bespelled
  • besplashed
  • bespotted
  • besprinkled
  • bestained
  • bestarred
  • bestormed
  • bestreaked
  • bestriped
  • bestudded
  • betailed
  • betasselled
  • betattered
  • bethumbed
  • bethumped
  • betimbered
  • betitled
  • betoiled
  • betossed
  • betrampled
  • bewandered
  • beweltered
  • bewigged
  • bewrapped

Solution 2:

It seems like a joke on the once common "bewhiskered". It is funnier if you are aware that "bewhiskered" does not only have a literal meaning, but can mean old, out of date, hoary. Note that it's a 26-year-old that's "bemustached".

Solution 3:

The English prefix be- was of course once productive; in some related languages such as German it still is to some extent. As the existence of the word bespectacled shows, the time when be- became unproductive in English wasn't even so long ago. As a result, when you treat it as productive today you will still be understood; you will also get an archaic and/or comical effect.

In addition, non-native speakers and excessive readers of old literature may have the mistaken impression that be- is still productive, or if they don't, might decide to treat it as if it were -- as an idiosyncrasy.