Alright new Jason Todd headcanons in a dpxdc setting:

Danny is a "liminal" ghost, rather than a "half" ghost. He's alive and dead at the same time. (He's like Jesus Christ (in the church denomination I grew up in), fully ghost and fully human.) Danny, in human form, can go through a ghost shield, because he IS a living human.

Jason, however, is a reanimated corpse. He isn't a ghost, wouldn't have a ghost core, etc, he has a normal human system that runs ON ectoplasm. Jason CANNOT go through a ghost shield, because he is always an ectoplasmic entity. Danny can go through the Fenton Ghost Catcher and be split into a ghost and a human; if Jason went through the ghost catcher, he would straight up die.

(For my purposes I'm gonna say that Jason became an ectoplasmic entity upon his resurrection, but wasn't very stable. Dunking in the Lazarus pit stabilized his system but also poisoned his ectoplasm.)

I do think that Jason could learn certain ghost abilities if he learned to harness his ectoplasm, especially if they detoxed him off the Lazarus waters. He's probably already enhancing his stealth and strength in ways he hasn't really noticed. I think he's held back by the amount of physical matter he's lugging around, so maybe he couldn't fly, but I'm imagining temporary invisibility, or intagibility of like, a limb at a time. Maybe he can't walk through walls, but in a fight he can dodge by instinctively making the targeted part of his body intangible.