Tenant request for eviction notice.

Possession - Section 8 and Section 21

My tenant who is receiving Universal Credit wants me to serve her with section 21 notice. Her aim is to get a council property  as she will be classified as  homeless with two young children, when she takes the section 21 notice to them. From past experience, I know that she will be told to stay there until I sought possesion through the court. And as we all know she will still be there until I go back to apply for the bailiffs to carry out the eviction. All this will be at a cost. 

My question is, can I ask for a rent increase to help me meet the cost of the eviction, as the process is most likely going to take longer than 3 months? The rent has never been increased since she started living in the flat in 2019.

Any comments and the right way to go about this will be highly appreciated.