Long shares sold directly by the trader (or shares bought to cover a short position) default to FIFO unless the trader designates to his broker what shares are to be sold at the time of the trade.
Prior to expiration, through a random process utilizing a "wheel", the OCC determines who will be assigned. That is then passed onto the broker who may have an internal process who will be assigned. At expiration, all options expiring one cent in-the-money will be exercised and assigned (Exercise By Exception").
I question whether one can intercede in this process. The only way to find out is to call your broker and ask if you have the ability to designate which shares will be assigned.
The obvious way to avoid all of this is prepare two orders, cover the call and almost simultaneously (seconds later) sell the shares that you want sold (designation). This isn't the best way exit because the market can move against you in that brief time span and you'll take a small haircut on the B/A spread. A Buy/Write combo order (sell to close) is the preferable route but can shares be designated with a closing combo order? Again, ask your broker.