Hi @Loroz,
I remember the submission times/deadlines being an issue when we used Turnitin with Canvas via LTI integration years ago. We now use the "Canvas plagiarism framework" integration instead and find that it generally works much better. There are some definite differences between the integrations and what may be available, but I'd recommend talking with your school/institution Canvas team about the possibilities of switching integrations to get around this issue (which I believe, ultimately is on the Turnitin side of things).
I hope this helps, even though it may not be an immediate fix.
-Chris