I want to say "klingons fight in order to be seen".

Suppose I write:

{tlhInganpu' luleghlu'meH Suv tlhInganpu'}

Would it be correct ?

What puzzles me is this..

The way "I've gotten used to the {-meH}", is that the subject of the {-meH} phrase, is the subject of the phrase which follows it too.

But is that the case, or can the object of the {-meH} phrase, be the subject of the phrase which follows it ?

So, could I write {tlhIngan leghmeH romuluSngan, jach tlhIngan} for "the klingon shouts, in order that the romulan sees him" ?

No, the object and/or subject of a purpose clause play no independent role in the main sentence.

When a purpose clause modifies a noun, there isn't even any object or subject, e.g., ja'chuqmeH rojHom. It's treated as an indivisible noun phrase.

When a purpose clause modifies a verb, the clause just hangs off the front of the main clause without directly participating in it. jagh luHoHmeH jagh lunejtaH: the purpose clause is not in any way an object of the main clause.

tlhInganpu' luleghlu'meH Suv tlhInganpu' means In order that Klingons are seen, Klingons fight. tlhIngan leghmeH romuluSngan jach tlhIngan means In order that the Romulan sees the Klingon, the Klingon shouts.

-- 
SuStel
http://trimboli.name