I just checked the rate on Exp.UK and it is indeed the "Exp. Special Rate" - although there is nothing special about it as it is exactly the same as the adv. purchase rate on hilton.com

I consider booking the first night on hilton.com and the second night on Exp.UK at the "Special Rate" using the GBP 40 voucher.

For the first night I’m likely to receive a room on the exec. floor as HH Gold and for the second night the hotel will probably let me stay in the same room. Should I tell at check-in that I stay for two nights or simply check-in again on the second day and tell them that I already stay at room XXXX? I assume that I could jeopardise the upgr. for the first night by revealing that I have booked a second night on Exp.UK.