I don't wear military insignia from any military unit I did not serve in, and don't wear official uniform items with civilian clothing, unless they're allowed by regs (e.g. the all-weather coat, black leather gloves, etc.). I'm sure others will have different opinions, but to me that seems to be the most reasonable and respectful. Perhaps you might even inquire with your great-uncle about whether it is customary for non-BW to wear the BW cap badge and red hackle?

Might I also suggest that you consider purchasing a small shadowbow where you can both proudly display these items related to your uncle's service and protect them from deterioration?

Best of luck!

Cordially,

David