There is no such thing as a "family" HSA. HSAs are individual accounts, not joint accounts, even though the distributions can be applied to the medical expenses of spouses and dependents.
If you maximize your your wife's HSA contribution for 7 months of eligibility, the correct numbers are:
7/12 x ($7,750 + $1,000) = $5,104 to your wife's HSA
5/12 x $3,850 + $1,000 = $2,604 to your HSA
Total = $7,708
You get the full $1,000 catch-up while your wife only gets 7/12 of the $1,000 catch-up
