"1099-B sent by new broker does not list cost-basis as they don't have acquired times and FMV of the shares to report cost basis."
"As the selling through new broker does not mention which lot and acquired date and all the shares are long term, so I believe I can pick from any acquired date and corresponding price."
Even if the new broker had all the information that the old broker had, the 1099-B would not list the FMV of the shares sold. The broker would report your discounted "out of pocket" cost, and that's still needed to properly report the sale, even if the sale is a Qualified disposition. The sale can still create reportable compensation, compensation that needs to be reported as and added to the out of pocket cost of the shares.
The default costing method in the absence of instructions to the contrary to the broker before settlement is FIFO so you're not really allowed to pick and choose lots after the fact.
If the old broker does have all the necessary information I'd try to get the correct lot information from them.