Do I need a License to Let

Absolutely not for the following reasons

The lease is where you start.
Does your lease say the following?
 

(8)      (a)      not to assign underlet or part with the possession of part only of the Premises
          (b)      not to underlet the whole of the Premises without the prior consent of the Lessor such consent not to be unreasonably withheld

 

(8) (a) means you cannot let only part of the Premises. - Self-explanatory.

(8) (b) is quite specific, you must have consent from Management but consent not to be unreasonably withheld.

‘prior consent’

Consent to underlet (not a Formal License to underlet).
The lease does not specify how the consent must be given. In fact it does not even specify that it must be in writing. Obviously to avoid argument later it is best to have the consent in writing, and a simple letter would suffice. (Email consent was given by Management in the past).
 

Such ‘consent not to be unreasonably withheld’

The following is direction from First-tier Tribunal (Property Chamber).
A “reasonable period” of time is often of primary importance, because even if a landlord intends to give permission, or is moving towards giving permission, he must nevertheless do so within a reasonable period of time, and if he takes too long the tenant can go ahead and assign regardless. Time runs from the date of service of the tenant’s application for consent, and there is no definition of how long a period is reasonable. It can vary on a case by case basis, depending on the context. The guidance of the Court of Appeal, per Mundy J (as he then was) is that “it may be that a reasonable time referred to in section 1(3) of Landlord and Tenant Act 1988 will sometimes have to be measured in weeks rather than days; but even in complicated cases, it should in my view be measured in weeks rather than months.” – Go West v Spigarolo [2003] EWCA Civ 17 at para.73

Cost of consent to sublet.
There is no provision in the lease for a charge to be made for consent to underlet.