Even a battery with the correct voltage may not be producing the correct amperage. Your first step would be to pull the battery and have it tested against its CCA rating.
I've seen the 1681 code on both of my skis all the time. My skis are 1st year 4-tecs, and I've read in the manual that the 1681 code can appear when the information center can't keep up with the data transfer rate of the MPEM. However, this code never prevented the ski from starting.
In the spring of 2018, one of my information centers was dead, but the ski would start. I was getting false DTCs too. I ended up replacing the part at a cost of $602.00. There are steps in the service manual to perform before you replace the information center, but it sounds like you've performed them ... check battery, check fuses.