Bounties for White Mantle Inquisitors are offered in addition to the standard daily bounty. These bounties can be accepted at any time (after accessing Lion's Arch Keep), but they can only be completed once per character. They count towards the 3 quest limit.
Bug! Like with Zaishen Challenge Quests, if you log out with a character immediately after turning in a completed Shining Blade quest, and do not log back in until that same quest becomes available again, you will not be able to take the new quest with that character. This can be avoided by zoning before you log out after turning in the completed quest; or of course logging in more often.
The day of the week that each bounty is available will always be the same because the total number of bounties, 21, is divisible by 7.