There are considerations beyond coding!
You may want to incentivize your syndicate partners by adding the partner's name next to the author's name. IE John Smith - A Dickies.com customer. The marketing considerations, not coding, are the more points of contact or more times a customer sees the brand name the more comfortable they become with the brand name and the more likely they become a purchaser of the product.
Dickies, like every other business benefits when people see their name on another site.
Another marketing question is how beneficial is a review or testimonial like ...
"The Acme mouse is the best mouse I've ever used" - Anonymous.
Certainly a review / testimonial like ...
"The Acme mouse is the best mouse I've ever used" - Bill Gates
... would be the one you want to promote.
Certainly your affiliate would like to see published on your site.
"The Acme mouse is the best mouse I've ever used" - Bill Gates - A Dickies Customer.
So I believe in this case the marketing department needs to be consulted.
Will google penalize me for this ?
If they don't doesn't mean that they will continue to not. But I don't know if I'd call it a penalty to ignore those reviews, which they may already be doing when there are no other reviews they can use.
Some of the fields are optional but listed as required
Many of the fields that generate a warning or even an error do not cause the schema to be rejected. For example, a street address for events is not considered optional - but in many rural locations where there are parks for example that don't have an address the event is used even though it is not complete or that is to say the field at the corner does not have a mail box.
Are they penalizing when the name is not there?
If they are just demoting the reviews for incomplete records, that may be their solution which is not a penalty in the broad sense but not promoting the "Acme mouse is the best mouse ... " - Anonymous.
If it were me going to marketing, I would suggest that when there are a lot of reviews for a product we only use the reviews with names, and if the reviewer has a title like, CEO of ACME Inc, they get listed first, and are sticky for at least several weeks, (we spread the goodwill). If the product does not have many reviews we post the reviews from Anonymous. But if marketing says they would rather not have any reviews from Anonymous, as the brand reputation is not based on anonymous people then go with that.