There's no harm in recording your serial number and reporting it to any register available. Check with your local police to see if they have something they encourage. For example, here in Brisbane there's the West End Serial Hub run by QPS.

I've previously used Bikesure who seemed ok.

If you have contents insurance, have a look into the PDS. Some insurers will cover it, but not while riding, while a number of others will cover it including while riding, as long as you're not participating in races.

Covering it under contents insurance is likely much cheaper than specialist bike insurers.

I have mine on the contents insurance. I've itemized the bike, DSLR, iPad, phone and laptop. I'm insured for $5k of stuff on my person away from home. I think I'm covered for a smaller amount (around $2k?) of stuff not listed above.

It was a little extra, but not an obscene amount.

I've thought about it but I have a sneaking suspicion an insurer would reject/complicate a claim based on details of compliance of the ebike to local regulations.