Last year we got the train from London to Dover. It was very straightforward and a short taxi ride from Dover station to the port. Train leaves from St Pancras station.
Whilst it is correct that it is normally straightforward from St Pancrus, currently there are no trains between Folkstone and Dover. So high speed trains terminate at Folkstone and you can book a taxi there. Suburban services from Victoria still go from Dover Priory but on a different part of the rail loop then the St Pancrus services.
High speed trains to Folkstone leave from the top right of StP, platforms 11 and above.
Other advice, if anyone is considering using London bridge station, do not, it's currently undergoing a massive refurbishment and best to avoid and many trains do not currently stop there.
Dover priory to port is a short taxi ride with taxis outside the station. Folkstone isn't that much further, just over the cliffs to the next town and a slightly longer cab journey.
As OP is staying in the Park Lane / Mayfair area, I suggest use of the Train via Victoria, get a cab to Victoria then the train to Dover, cutting out the rail works at Folkstone.
Care some trains if say 8/12 carriages split around Faversham, Kent, so ensure your in the correct part of the train. Front/ Back.
Note the line indicated in Black, doesn't go to Dover this summer, and past Ashford, terminates at Folkstone.
https://www.londontoolkit.com/travel/dover_train.htm
Current Dover forecast-
Not looking great for Saturday, we had storms come in today, very cold and windy, and looks like it may take until Sunday to recover to having some sunshine. - sorry.