In the case of a melanoma affecting an eyelid, I don't think that it is necessary to have an oncological surgeon remove the tumor. However, it is best to have someone remove the tumor who is going to be aggressive enough to remove a wide margin, which may make it necessary to repair the lid margin using a corrective surgical procedure. So I do think it might be worth discussing this with your vet. If he is comfortable doing this kind of surgery then there is no problem. If he is not, I would think an ophthalmologist or a board certified surgeon would both be good alternatives, although both would probably charge more than a general practitioner for the surgical procedures.
Radiation therapy can be helpful.
Removal of the eye, if it seems as if it might contain the tumor, is a reasonable option. I cannot find any info on your website. These do not usually recur after surgical removal if a wide incision is made and tissue removed under the melanoma, too. It is a good idea to have tumors suspected of being melanomas removed and examined by a pathologist, who can give some idea of the potential malignancy. If it seem appropriate after examination by the pathologist chemotherapy can be used to try to limit the spread of the melanoma.