A possible solution for some places might be one of the small two-person carts.

These somewhat resemble an electric golf cart, but are much smaller.

While the wider stance of the cart would make it unsuitable for many congested areas, there are still a lot of places where one would permit access without creating a hazard to others. (You would need to check in advance with museums and similar places to make sure that narrow aisles or narrow-minded policies don't restrict them.)

With one of these, father would not be "displaying his own disability," he would be providing transportation for her (if she's willing to let him drive).

If his use of an assist could be presented as his "being there to assist her" perhaps he might even be more willing to accept use of a separate scooter or motor chair "because of her need" to have him nearby during excursions.

While this approach might seem a bit "manipulative" it would appear to be a valid concept that could be presented for his consideration(?).

Having used a "walking staff" for decades, since long before I had any limitations on basic mobility, I might suggest that a trip to a larger sporting goods store to show him that "alpine staffs" (similar to ski poles) are used by athletes in "speed hiking" and for more general assistance in moderately rough country might change his opinion on more basic "aids." (A mobility aid doesn't have to look "medical" to be effective.)

The key may just be to avoid treating the aids as "something he needs" and instead calling them something that will let him help someone else more effectively. That he does have someone to "depend on him" perhaps offers an opportunity to try this approach.

John