I have received two other suggested reroutes around this closure.
1) Aguajito Rd.
Northbound: Exit SR 1 at #399A/SR 68, before entering SR 68, turn right onto Aguajito Rd. In approximately 4 miles, Aguajito Rd. ends at a signal. Turn left towards downtown on Fremont St. and rejoin route at Abrego St.
Southbound: From Abrego St., turn left onto Fremont St. Before it enters SR 1, turn right toward Fairgrounds onto unsigned Agujito St. In approximately 4 miles, Aguajito St. ends. Turn right, ride over SR 68. Turn left and enter SR 1 SB and rejoin route.
2) Follow the coastline.
Info from Monterey County: Cyclists could take a long, but flat and scenic route along the coastline of the Monterey Peninsula by riding the Monterey Bay Coastal trail along the coastline through Monterey and Pacific Grove to Ocean View Blvd, following Ocean View Blvd along the coastline through Pacific Grove, and using 17 Mile Drive through Pebble Beach to Carmel-by-the-Sea. The segment of 17 Mile Drive between the Lodge at Pebble Beach and the Carmel by-the-Sea city limits is narrow, with a good amount of vehicle traffic, so may or may not be advisable for some cyclists.