[RPG] Can a Paladin use his Mercies with Channel Positive Energy

Not by RAW, no

Though the Channel Energy ability consumes uses of Lay on Hands, it is not Lay on Hands. As a result, the bracers do not affect Channel Energy directly. However, they will still grant additional uses of Lay on Hands, which you're free to spend howsoever you wish (such as, say, by fueling Channel Energy).

Funnily enough, works by RAW.

There is no prohibition of magical healing in construct traits, though some constructs are explicitly immune to magic except select few effects.

There is also no prohibition in the heal spell.

Technically, you can do it, but expect a heavy book thrown by your DM. He will also probably houserule it away, at least I personally would.

Note:

Can't remember anything on that in 3.5 either, though the description of Warforged contrasts the possibility of both special and positive-energy healing (emphasis mine):

As living constructs, warforged can be affected by spells that target living creatures as well as by those that target constructs. Damage dealt to a warforged can be healed by a cure light wounds spell or a repair light damage spell, for example, and a warforged is vulnerable to disable construct and harm. However, spells from the healing subschool and supernatural abilities that cure hit point damage or ability damage provide only half their normal effect to a warforged.

That leads one to the implication of impossibility of positive-energy based healing for a regular construct, even if it is not stated anywhere.