But "it's not important" is Bullshit.
As his wife, if he's ill or in an accident, she's next of kin to visit in wards or make medical decisions while incapacitated.
She's due any widow payment from the government even if you're arranging the funeral.
She's the assumed beneficiary if he has spouse benefits on his pension.
Inheritance tax makes their joint estate tax advantageous, NOT anything you build and leave for each other in a will.
Joint finances when it comes to credit reporting or debts - won't even begin to mention the horror stories common in that area, I'm sure some are obvious.
OP, I'd be out like a shot and tell him why - it IS important that he legally reflects the marriage breakdown, not least to protect other people he does supposedly care about.
And I'm only looking at this from a financial/basic legal POV.. I'll let others point out any moral, religious etc issues (as I think practicalities are universal, the other stuff is more down to our personal outlook).