The divorce will not affect your LPR status. You will still need to file an I-90 for a renewal of your card before it expires, but that won't be until 2027. Or you can naturalize, in which case you won't need a greed card anymore.
There's no such thing as registering a divorce in the US. The foreign divorce documents are accepted.
He can threaten whatever he wants, but he has no say in the process. It's actually foolish of him to try to stop you...he's on the hook for you via the I-864 he signed still. If/When you become a US citizen, his I-864 obligations go away. It's to his benefit if you become a citizen.
