It appears that there is no readily available source for this claim.
Other than claims beginning with the words "It is thought that..." or "It is believed that," I could not find any reputable source for this belief (actually, I couldn't find any source at all except for the hearsay-sounding examples above).
TL;DR: I searched Google Books and Amazon for instances of "Bernard Corleone" and "Philip Latini" (pre-monk name) and didn't find any instance of the claim. Below are screen shots of my findings and the summary at the very end lists my interpretation of this lack of mention.
Sklivvz has already pointed out in his answer that the official Vatican biography does not include the claim. Here are some others which fail to reference it:
This book features a preview on Google Books allowing the full read of Bernard of Corleone. The entry is on Page 14 (the book is organized by feast date; his entry is January 12th). Here is the screenshot of the end of the entry, which makes no note of this claim (or even the apparition of Mary placing the baby Jesus in his arms; the only apparition mentioned is from Jesus himself):
The preview on Google Books contains no mention:
The Google Books preview contains no mention:
The massive 12 volume Butler's Lives of the Saints January Volume only mentions him as a footnote (from Amazon preview):
Bernard Bangley, *Butler's Lives of the Saints, Concise and Modernized Edition
Features this entry extracted from the Amazon preview, and makes no mention:
Summary: While it's impossible to prove the statement, "Bernard of Corleone did not know the date of his death" or "No evidence exists that Bernard of Corleone knew the date of his death," I find the above compelling enough to think that there is probably not readily available and reliable evidence to support the claim.
The fact that the first book recounts a vision of Jesus gives me reason to think that if there was also evidence to support the foreknowledge of his death, it would have been included. I can't know this for sure. All I know is that five comprehensive books aimed at presenting the lives of the Saints in an inspirational manner did not list this would-be-inspiring tidbit for its intended audience (probably Catholics).
As for my method, I searched Google Books and Amazon for "Bernard Corleone" and "Philip Latini" (his pre-Capuchin monk name) and looked at pretty much all the books I could find. There were some older versions of Butler's Lives of the Saints that mentioned the name, but Google didn't have the full entry available. Current versions of Butler's Lives of the Saints, as shown above, either don't mention it or don't list him except as a footnote.