Same issues as the OP with my own Wells Fargo Debit Card seamlessly working with my iPhone 7+ Wallet, but not passing the verification process to my Series 2 Apple Watch Wallet.
I found the Wells Fargo App has a settings area for adding your cards into Wallet for you (as opposed to adding the cards into Wallet directly) **with a distinction between the phone and watch**.
When I chose the phone option first I made it through the process successfully and afterwards found the status of the card simply: 'Added' with the watch option no longer available... and the the main issue of not being able to the pass it along to my watch continued on.
However.. when I reversed the order and did the Watch First it successfully went through! Note that the debit card had NOT been added to the Phone's Wallet at this point, but with the watch having now successfully verified the card, I found the WF App proceeded to let me then verify it on the phone as well and suddenly everything is working as intended!! It seems totally backwards to add it to the Watch first but I guess WF had to make it messy..
TLDR:
1. Delete WF Card from phone wallet.
2. Use WF App to add Card to Apple Watch first.
3. Use WF App to add Card to iPhone second.
4. Live Long And Prosper.
PS - This is obviously a possible solution for those with a Wells Fargo Debit Card, if you are having this trouble with another bank's card then I would try to see if their app can setup the cards in both phone and watch wallet for you.