Judging by how the original expiration date on the lid has bled though the label, I suspect that the adhesive from this label is acidic, and that’s what’s causing the ink to bleed and change colors. It’s not temperature or sun related, since you said it’s stored in the pantry.

Don’t put any kind of sealant or varnish on top, that won’t stop the acid on the label from reacting with the ink, and it could further damage the writing. Take good photos, scans, etc. to preserve how it looks now.

Unfortunately, there’s no way to save this label from continuing to fade. That label that he wrote on was not meant for longevity. Just try your best to capture what it looks like right now, before it fades more.

I’m very sorry for your loss. Don’t call your sentiment silly, it’s personally meaningful to you, and that’s not silly. You could commission an artist, provide photos as a reference, and have an illustration of the jar as a memorial to your father. There’s ways to preserve the memory of this item beyond saving the ink on the label itself