Thanks for your question!
We offer a range of weatherproof label materials that are very durable and water-resistant at the link below. They won’t break down when exposed to oil, but direct contact may cause the print to smudge or discolor.
www.onlinelabels.com/materials/weatherproof-labels
The HP 3201dw is a solid choice, and we’ve also seen great results with the HP LaserJet Pro series overall. While we haven’t tested every model in-house, many popular printers from HP, Epson, and Canon work well with our materials, and customers have reported good results too.
When choosing a printer, be sure to check that it supports specialty label materials, specifically film materials in your case, and review the printable area—some models have non-printable margins that can affect edge-to-edge printing. You can read more about that
here.
For best results, use a “Labels,” “Cardstock,” or “Transparency” setting. For inkjet users, adding a laminate or clear gloss overlay can help protect against oil.
If you need maximum oil resistance, our custom roll labels are laminated after printing and are a great option:
www.onlinelabels.com/custom-roll-labels
Hope this helps!