A pwd can have capacity in regard to some matters and not have capacity in regard to others. With my mother, it depends on how she is on the day, which in turn seems to depend on what she's been doing in the preceding few days sometimes and at others heaven only knows.
Neither my mother's previous bank, nor her current bank have made any enquiries at all into her mental capacity; I just opened/closed and run the accounts under the LPA. The same applies to the accounts in which we hold the proceeds of sale of her house. When her house was sold, the solicitor said either she'd got to meet her to check capacity to make sure the sale is valid if Mum signed it, or we'd sell it under the LPA and I could sign it off and they wouldn't worry about Mum's capacity. So, I'm not sure the bank accepting the registered lpa so that you can act is going to help a great deal, although I suppose it may be a pointer.
When it comes to changing the will, if a solicitor is drawing it up, he must be satisfied that the testator has capacity to understand what he's doing. The sons might be able to swing it I suppose - but a lawyer worth his salt, should make absolutely sure they were not influencing him. The sons could write the new will themselves and get their father to sign it. As I understand it, if you are were to cite "undue influence" as a reason for disputing a new will, the burden of proof is on you. I'm not sure where the burden of proof lies if your challenge is based on lack of capacity to sign - I suspect the same may apply. In short, if you have doubts, and only you will fully understand the dynamics here, then yes, get a couple of professionals to confirm he doesn't have capacity to make a will so that, if the sons do try and get your husband to re-write his will, you have the papers to challenge rather than having to go back to people for evidence possibly years down the line.
All that said, I've come across this sort of thing at work, but I'm long retired and so rather out of it all nowadays. To be on the safe side, I'd consult a solicitor if I were you. You could just pay for an hour or so to talk through the problem with someone suitably experienced in the subject.