Touristic visit during cooling off period

Hello everybody!

I am currently in the UK on a Tier 5 Visa as a Temporary Worker. My visa expires in July and I'll leave go back to Italy by then. I intend to apply again for this visa after the 12-month cooling-off period. However, I was wondering whether, during the cooling-off period, I could come back to the UK for one week just as a standard visitor (maybe a visit for touristic purposes in August).
In that case, could I enter the country even if my BRP has expired? And, if I come, would that compromise my cooling-off period so that next year I won't be able to apply again?