The foreigners mourned with perpetual lamentations for 3 months following Murray's death.
Previous answers seem to focus on the literal meaning of "perpetual". But the Oxford dictionary gives this secondary definition:
Occurring repeatedly; so frequent as to seem endless and uninterrupted.
I would say then that your sentence is quite acceptable. As a native British English speaker I can confirm that "perpetual" is used in a looser, hyperbolic way to mean either frequent or persistent, for example:
Stop your perpetual whining
A "perpetual whiner" is quite a common insult (Google it!) and obviously does not mean that someone complains 24 hours a day every day without letup. It just means that they frequently complain.
Likewise, your sentence does not literally mean that their lamenting is expected to go on forever, just that it is so frequent and persistent so as to seem that way.
It does not make sense, as perpetual means "without end, forever" and 3 months is a limited time frame. Perpetual is a stronger word than even "endless", which is a word that would work in this sentence. "Endless lamentations for 3 months..." would imply that they didn't stop their lamentations during these 3 months, but did stop after the 3 months. Perpetual is a word that is exclusively used for things that NEVER end.
As pointed by my namesake, Richard, perpetual is not the right word to be used in your phrase.
According to the Cambridge Dictionary
continuing forever, or happening all the time
for all time
For all time, during an unlimited time, not just for 3 months. Maybe the word that you are looking for is uninterruptedly
in a way that is without any pauses or interruptions
The foreigners mourned uninterruptedly for 3 months following Murray's death