No one has reported being denied a fastpass exchange card at any time of day, even at the busiest times of year. We redeemed ours at Epcot at 6 pm with no trouble, but we redeemed at DHS at 9:30. We intentionally came after DHS rope drop so we wouldn't get caught in the TSM morning stampede. Even if TSM FPs are gone (which they probably will be by lunchtime), you will still be able to get FPs with the GAD FP exchange cards.
Disney states that FP exchange cards may be limited, but no one has reported on the DIS that they couldn't get them. Disney also states that they may not be valid during extra magic hours, so you probably can't redeem during evening extra magic hours (in case you meant that late).