Hi. I've had this exact same issue. I do have a fix of sorts. Deleting and reinstalling the iCloud account does work BUT!! this then deletes all the sent messages in the SENT folder on Outlook which haven't synced. I found this to my cost. The fix described previous on this thread, to set up a new iCloud account, but not using microsoft sync services does seem to address the issue, but noting my previous comment, if this is a replacement account you may lose sent emails in Outlook.
My fix is to set up a second iCloud account using the alternative iCloud address, and don't use sync services. Outlook won't let you set up a duplicate with the same email address, but the alternative XXX@icloud ******@mac.com or ******@me.com seems to work.
UPDATE... Meh. It looks like the new account (not using microsoft sync) doesn't allow search facility - its might be that its re-indexing but at present I can't search for emails in the 'new' account.
UPDATE 2 .. OK it worked for a while, but then sync failed again, so back to square 1.
Would be nice to have a proper fix for this...