Stealing ideas from Sleafar's first solution, I got to 62 moves. So hopefully someone can better that with the magic 64 or more.

The king moves from h1 to h3.
1: Capture c8 and then a7,a8,b7,a6 (14 moves)
2: Block the queen's line with the free pawn (3 moves)
3: Capture a1 and then the now unprotected d4 (16 moves)
4: Free b3 for promotion by capturing c6,b5,d5,b4,b6 (10 moves)
5: Move to e.g. g4 (5 moves)
6: promote (to queen or rook) and clear h8 (8 moves)
7: advance the pawns and move to the target square (6 moves)
Total: 14+3+16+10+5+8+6 = 62 moves

Stealing ideas from Sleafar's first solution, I got to 62 moves. So hopefully someone can better that with the magic 64 or more.
EDIT: Since 62 is beaten by Sleafar, I'll up my score to 70: Earlier I failed to realize a8 could just be a pawn; freeing up a bishop

h1 must be removed to reach a1. Doing that cost 8 moves, after which 62 moves are needed exactly like in my first solution. The top left cluster could be removed before capturing h1, but that also results in 70 moves total. Note that the pawn cannot block the bishop's line, since it is needed to block the queen's line. The king moves from h1 to h3.
1: Capture c8 and then a7,a8,b7,a6 (14 moves)
2: Block the queen's line with the free pawn (3 moves)
3: Capture a1 and then the now unprotected d4 (16 moves)
4: Free b3 for promotion by capturing c6,b5,d5,b4,b6 (10 moves)
5: Move to e.g. g4 (5 moves)
6: promote (to queen or rook) and clear h8 (8 moves)
7: advance the pawns and move to the target square (6 moves)
Total: 14+3+16+10+5+8+6 = 62 moves

Stealing ideas from Sleafar's first solution, I got to 62 moves. So hopefully someone can better that with the magic 64 or more.

The king moves from h1 to h3.
1: Capture c8 and then a6,a8,b7,a6 (14 moves)
2: Block the queen's line with the free pawn (3 moves)
3: Capture a1 and then the now unprotected d4 (16 moves)
4: Free b3 for promotion by capturing c6,b5,d5,b4,b6 (10 moves)
5: Move to e.g. g4 (5 moves)
6: promote (to queen or rook) and clear h8 (8 moves)
7: advance the pawns and move to the target square (6 moves)
Total: 14+3+16+10+5+8+6 = 62 moves

Stealing ideas from Sleafar's first solution, I got to 62 moves. So hopefully someone can better that with the magic 64 or more.

The king moves from h1 to h3.
1: Capture c8 and then a7,a8,b7,a6 (14 moves)
2: Block the queen's line with the free pawn (3 moves)
3: Capture a1 and then the now unprotected d4 (16 moves)
4: Free b3 for promotion by capturing c6,b5,d5,b4,b6 (10 moves)
5: Move to e.g. g4 (5 moves)
6: promote (to queen or rook) and clear h8 (8 moves)
7: advance the pawns and move to the target square (6 moves)
Total: 14+3+16+10+5+8+6 = 62 moves