My strong assumption is he would have been King Frederick. He is known in actuality as Frederick Prince of Wales, so presumably would have been king by that name. Likewise, I agree with Damian that Ernest would have been King Ernest.
There is no tradition of monarchs with double names in Britain, which has a lot to do with why Albert Edward dropped one of his names upon accession. Likewise Alexandrina Victoria, although she was no longer known by both names in casual usage by the time of her accession.
Unlike Damian, I do not think there is much chance that Frederick or Ernest would have taken the name of a predecessor. That was really only ever done by George VI and Robert III of Scotland back in medieval times, and those were both under exceptional circumstances.
Is it ever known under which name would he have reigned ?
Frederick I ?
Frederick Lewis I ?
The UK Had never had a double named king before.
I'm guessing he would either have reigned as King Frederick (no ordinal number would be used until a second King Frederick occurred) or taken his father's name or the name of a previous British monarch.
If the Duke of Kent had not fathered Victoria, there might have been a King Ernest-August I of the UK.
As with his grandfather above, he would have reigned either as King Ernest or taken the name of a predecessor.
In the 20th century, three kings assumed regnal names in place of their given names ie. Albert Edward became Edward VII, David became Edward VIII, Albert became George VI so that may well have been the same tradition followed by those earlier examples.