Can I request small claims court

Hi all, I have a court hearing in a few weeks time which I assume is the direction hearing. My tenant has provided a defence as they want to stay until they get a council house which can take years but they also want to pay well below market rate.
Tenants legal rep has sent me an email with draft directions. They will propose it to go to fast track claim court. What they have proposed adds further 6 months or more with inspection of the property and so on.
In the meantime tenant is still not paying the correct amount or on time.
My question is can I request the judge to consider small claims court. This is usually where the judge bases his decision on the written evidence provided which I have lots of evidence of. Also from what I have read it doesn’t take as long. Also another thing I have read is with small claims court, even if me as the claimant loses the judge doesn’t usually expect us to pay the costs but I could be wrong.
Any insight in on this would be appreciated.

Forgot to add the rent arrears are about £4K as I know small claims court is for figures under £10k. However I’m not fussed about the arrears I just want possession of the property now.