Can my wholesale group get a discount on one category only?

Yes, with a category rule on the group.

1. Go to Customers → Customer Groups and click Edit on the group.
2. Under Discount rules, click + Add rule.
3. Choose the Rule type: Percentage or Fixed amount.
4. Under Applies to, pick Specific categories and select the categories.
5. Enter the Percent % (or Amount). Min cart and Max cap are optional.
6. Click Add rule, then Save group.

The rule only applies to signed-in customers in that group. You can add several rules to the same group, for example a different percentage on another category.