Per The Club website under the FAQ:

"Who can use the lounge?

Our lounges are located after security in the departure area and are available to checked-in departing or arriving passengers (please note this excludes arriving passengers at The Club CHS and Club Aspire Heathrow Terminal 3 & 5). It doesn't matter which airline or class of ticket you've got, you're welcome to use The Club and Club Aspire lounges.
If you have a lounge access membership card, members of Priority Pass, Lounge Key and Lounge Club are accepted in all our lounges. Access is subject to availability without a reservation."

I haven't had issues using on arrival at other The Club locations since they made this change a couple years ago and sounds like you have had success before, too. I would just show them The Club website stating access. https://theclubairportlounges.com/co...tional-airport