Problems syncing catalogues in Lightroom from my old MacBook Pro to my new iMac

I appears you have an older version of Lightroom on your Mojave Mac and a newer version on your new iMac. The new version upgrades your old catalog in order to use it. But you can't go back because the old version doesn't support the new version's catalog. So, 2-way "sync" isn't possible in this case.

It may help if you can tell us exactly what version of macOS & Lightroom is on each Mac.

Like very many apps, Lightroom has never been backward compatible. LR 13 would have never been compatible with an LR 10 catalog, it would have always required the catalog to be upgraded. Are you saying that you previously had LR 10 on your M1 iMac and that you upgraded it to LR 13?

What version of macOS are you running on your M1 iMac?

As long as you have a CC subscription, you may be able to request & get a copy of LR 10 from Adobe. LR 10 was compatible with Mojave, Catalina & Big Sur. Whether or not it can be installed on Monterey or later is unknown (hopefully Adobe can answer that question). LR 13 is compatible with Monterey and presumably Ventura & Sonoma.

Have you considered upgrading your MBP to Monterey, in which case you could run LR 13 on both Macs.

You're welcome. I hope it works out.

I agree about the expense of buying new computers and do as much as I can to take care of my Macs and keep them in good running order for the long term. In fact, I'm posting this from a mid-2012 MBP that is my go-to Mac when I'm travelling. It's still going strong. Still running High Sierra, which has been extremely stable, and am considering putting in an SSD and upgrading to Catalina for a few more years' use.