Fairly tricky this one.
Two options would be:
1) Loe Beach in Feock (not THAT close to Falmouth)
2) Polgwidden Cove in Mawnen Smith (only accessible through Trebah Gardens)
Both have no dog restrictions as far as I'm aware but both have drawbacks.
Otherwise, I think you are looking at waiting until October 1st but I'm happy to be corrected!
25k+ basic/com
21000.00 - 24000.00 pa
20,207.80 per annum