We all know the din that if you miss a day of sefira you can no longer count with a Beracha (this is the shita of the Behag). However, if you remember during the day you count without a beracha and then can continue counting with a beracha. L'halacha the acharonim say we do this because of a s'fek s'feka. However, the Behag himself who held that you are not יוצא during the day said this. The obvious question is that if you are not יוצא (you don't make a beracha) why can you continue the next night with a beracha?
RYBS explained as follows (Mesora volume 3). The Behag said that if you miss a day you can no longer count with a beracha because of a lack of תמימות. What does this mean? RYBS explained that תמימות means continuity. If you are counting you cannot skip a number, you can't count 1,2,3,4,6.
Based on this we can understand why the Behag says to count during the day without a beracha. Although you are not יוצא the mitzva, you did count the day. Therefore, that night when you get up to count you have the continuity of the count.
He brought a number of proofs to this, I will mention 1 of them.
R' Hai Gaon (quoted by the Mishna Berura) says that if you forgot a day, the next night before you count you should first say that yesterday was so and so. The obvious question is why? Yesterday is over, what good does it to to say today that yesterday was 6? The answer is that R' Hai held like the Behag, that counting needs continuity, Therefore if you mention that yesterday was so and so you can then continue to count today, as you have the continuity.
Based on this, a child who becomes Bar Mitzva during sefira would certainly be able to counting as even if we consider that what he did as a minor was not a real מעשה מצוה it doesn't matter, he certainly did count. Therefore, just like someone who counts during the day can continue even though they were not מקיים the mitzva, so too a the Bar Mitzva boy can continue counting with a beracha.