If I were you, I’d just avoid working with this buyer. However, to answer the point in question, if you deliver by uploading a file and sending it via the Fiverr delivery system, you are not classed as working for hire.
I offer writing. Where I am based (and in most other tax jurisdictions, I am obliged to pay VAT on any document I physically or electronically deliver to a client. This is because I am creating and delivering something of value.
Conversely, if I offered to work with buyers onsite and typed an article directly onto their website rather than delivering a file, I would not be obliged to pay VAT on that order. This is because I am not delivering a physical item, I am providing a service and essentially “working for hire.”
Throw that at your buyer and see how they like it . (Don’t really.)
As for how this applies in practice to Fiverr TOS, the last time I looked, Fiverr does say that all work belongs to buyers on delivery. However, TOS also states that some sellers may charge commercial rights extras for some services.
While your buyer may decide to blow smoke through their ears and argue they don’t need to pay for commercial rights, it is likely that they would not pursue this matter in court (given your prices).
If I were you, I would simply avoid working with this buyer, as you don’t really know what they are trying to maneuver you into by acting the way they are. My guess would be that even if they do purchase necessary rights, they may come back later insisting that they now own the rights to the overall illustration style you have used in their project. As such, they may ask you for a refund (because you are obviously still offering similar style drawings to other buyers), or they may even insist that you stop offering your service altogether.
Refuse to work with them, cancel if they do order, and block them. That’s what I would do.