Is there a way to see the precise time of when a transaction was sent out?
While Murch answers the technical side of it already, what you're trying to do is simply not possible. There is no provision in Bitcoin to know the exact time a transaction was broadcast (short of running several nodes that connect to enough of the network to attempt to deduce the first broadcast time).
If you're trying to protect yourself against price fluctuations, your best bet would be to have a system that requires a certain number of confirmations within a time window of your choosing (say, 30-60 minutes). If you have not received BTC in that time period, you can cancel the order. If you receive it after the window has closed, you simply refund the BTC back to the customer and ask them to place a new order (or let them use that BTC amount as a credit against their new order).
This ensures that you only have to sell BTC for fiat within a certain window of time, and if the BTC transaction occurs too far after the order is placed, you return the BTC, without touching fiat at all.