Whether the emperor meant it as a "pun" (or something similar) is pretty hard to know. But it is doubtful.
昭和 means harmony for the same reason that 和 is associated to Japan: in both cases, 和 represents the very specifically-Japanese conception of "social harmony" (which is different enough from the western word, for many scholars to use "Wa", even in English texts). It is not a coincidence, but it predates our era by a good 13 centuries.
The use of 和 in the spelling of 大和【やまと】(old name for Japan) has a long and complex history, but the gist of it is that it was picked as a replacement for a previous kanji used by the Chinese (倭), which the Japanese did not find flattering. 和 did have a positive association with a philosophical concept that the Japanese considered at the heart of their culture, hence the entanglement and how it became to be both "harmony" and "japan".
To answer your question of whether it carried a meaning of Harmony specifically for Japan: yes, in a way, since "Wa" is the Japanese conception of harmony, and foreigners (particularly of this era) would definitely not be expected to grasp its meaning, let alone have it.