would be interested in this too as to me it falls in the “while it may not be ideal sometimes you gotta do what you gotta do” category and it is better to be warmed up rather than training cold. NOTE I am talking about walking speed actions, abs, dynamic stretches and sometimes drills.
Re the scarf, you should be ok if you don’t make it too tight or have a scarf that too thick. May not work but worth a crack. If you find you collapse due to a lack of oxygen then I suggest it isn’t worth proceeding 